- Email: [email protected]
Electron-phonon coupling on the ionic 60-atom carbon cluster buckminsterfullerene
J. Phys. C&m. Solids Vol. 54, No. 5. pp. X-573, Printul in Great Britain.
1993
oDz2-3#7/93 sf&i + 0.00 g’: 1993 Pergamon Press ttd
ELECTRON-PHONON COUPLING ON THE IONIC BO-ATOM CARBON CLUSTER BUCKMINSTERFULLERENE U-HYON PAEK$
KEE HAG LE@ and
@Iq3artrnent of Chemistry, ~o~wang
University, Iri, 570-749, Kwea ffkpartment of chemistry, Gyeongsang Nationaf University, Jinju 660-701, Korea (Receiued I May I992; accepted in revised j&m 12 November 1992) Abafract-Using a tight-binding model with electron-phonon coupling based on the modified SSH model Hamiltonian, we studied several ground states of the neutral and ionic 60-atom carbon cluster, buc~nsterfull~rene. The ground states of ionic structures are shown to consist of Jahn-Teller distortion and localized self-trapped electron states dependent upon the ratio of the electron-phonon coupling constant to the electron hopping constant. Keywords:
Ionic C,
ground state, tight-binding methods, electron-phonon
coupling, Jahn-Teller distor-
tion, self-localizat&
The recent synthesis of macroscopic quantities [I] of the C, fullerenes has made the discovery of the chemical and physical properties of this new molecular form of carbon possible. Spectroscopic [Z] and diffraction ]3] studies have confirmed the predicted truncated icosahedrai structure [4] of the C, molecule. Recently, the formation of new semiconducting (51, high conducting ]6] and superconducting [?j anionic charge transfer compounds of C, witb molecular and a&ah metal counterions, respeetive]y, has been discovered. Hofzer er trf &rj have found that the approximate composition of M3Cso (M = K, Rb) is the superconducting phase, and neither under- nor over-doped phases are superconducting. Phot~mission studies of K,C, emphasize changes in the valence bands, correctly noting the filling of the lowest unoccupied molecular orbitals (LUMOs) derived bands [8], and estimate their stoichiometries from the emission intensities of band features derived from the highest occupied molectdar orbitals (HOMOs) and the LWMOs. Renning et & [g] have suggested that this filling is not rigid band-hke and K,C, is a strong electron-phonon coupling superconductor. Poirier et al. [S] and Chen ef af. [9] reported that K,Cso samples with intermediate stoichiometries represented two-phase regions of an ~~l~b~urn phase diagrams. Chen et al. [9] also suggested that these systems are weakly coupled BCS-like superconductors. With synchrotron X-ray powder diffraction, McCauley ‘et al. [lO] reported that the miscibility gap in I&C, implied by photoemission results [9] did not cover the entire Otx<3 range.
Tanigaki et al. [q and Fleming e? al. [7j reported that the variation of ?Y,with dopant supports the interpretation that the transition temperature of these fullerides is determined mainly by the density of states at the Fermi level. Duclos et al. [I I] reported that the Raman fr~uen~es for doped M,C, (where M is K, Rb or Cs) films are very similar, indicating little change in intramolecular phonon modes. Irrespective of the details of the precise microscopic mechanism of superconductivity in alkali metal doped C,, the density of states at the Fermi level and electron--phonon interactions would play a crucial role in determining their high T,-values 17, 1I]. ft is therefore reasonable to assume that this would be consistent with the idea that C, intramolecular phonon modes in a single molecule are a very important factor of su~~onducti~ty in the doped Cso systems. This has already been suggested by !&hluter et al., Varma et al. and Anderson et al. [ 111.They find that tangential atomic vibrations (with C atoms moving on the sphere surface) coupled more strongly with electrons. In intramolecular phonon theories of ~~r~nducti~ty in I&C,, Schfuter ef ai. fl 1) and Varma et uf. [l l] have shown that electron scattering is dominated by particular on-ball Jahn-Teller-type modes on the scale of the large on-ball n-hopping energy. Here Varma et al. [l l] calculate the characteristic intramol~ular defo~ation potential per unit displacement of the specific mode which is based on the Jahn-Teller deformations of a C, molecule. Schluter ef al. [ 1l] shows that the third electron not only changes tbe magnitude of the Jahn-Teller distortion but also changes its form. Therefore, it is interesting to see the Jahn-Teller distortion potential
565
566
KEEHAG LEE and U-HYMNPAEK
of C& (where n = I, 2,3) as a function of the doping concentration. Wehner and Van Zee f12] and Kroto et al. 1123 have presented a comprehensive overview of the theoretical work on fullerene-60. Hayden and Mele [13] have studied several ground and excited states of the neutral and ionic C,, cluster using the tightbinding mode1 with electron-phonon coupling. They found that the singly-ionic ground state underwent Jahn-Teller distortion, but that no localized selftrapped electronic states occurred. Negri et al. [14] have classified Jahn-Teller instabilities in both excited neutral and ground ionic electronic states. Recentiy, Clementi et al. fi5] performed ab i&if3 calculations for the ground states of the neutral and ionic C, cluster within the constraint of ih geometry. As far as we are aware, the general effect of electron-phonon coupling in ionic Cm ground states is not known. It would be interesting to see if electrons in ionized ground states of reduced symmetry have bcalized or extended character in a system intermediate in size between small clusters and solids. The soliton model of Su-Shriffer-Heeger (SSH), which is the tight-binding mode1 with electron-phonon coupling, demonstrates the novel phenomena with the midgap state in ~1ya~tyIene f16]. In this paper, we present the results from self-consistent numerical calculations which allow for a complete relaxation of all n: electrons and individual atoms in the ground state of neutral and ionized C, systems. Our self-consistent, orthogonahzed linear ~rn~nati~n of atomic orbitals (0LCAO) method is based on the SSH Hamiltonian model, which hereafter is called the modified SSH model Hamiltonian. The variation of lattice deformation in each system is considered as a function of the ratio of the electron-phonon coupling constant (S) to the ~~~~on-hopping constant (T). The iocalized or extended character of the ekectronic states with the lattice deformation is also discussed. We construct the model Hamiltonian calculation scheme in section 2. The self-consistent computation, which is similar to the method proposed by Shastry flY& and the Schmidt o~hogona~~tion of alI 7corbitals in order to consider only the lattice vibration modes are used in our approach. The particular utility of our model calculations lies in the speed and ease with which they yield correlation diagrams. The criterion for convergermy was set to JO-‘* for the charge iteration to ensure a highly refined structure for all calculations. But for C& ionic ground states only in the range of 0.1-0.4 for s/t values, we obtained the convergency with lo-’ for the charge iteration. Ait our results will be compared to the findings of other authors.
2. MODEL HAMILTONIAN CALCULATION SCWEME As a model for neutral and ionized C, systems, we consider a buckminsterfullerene composed of 60 lattice sites and N electrons (N 3 60). Each site has one electronic state per spin direction corresponding to the carbon x orbital of OLCAO in the neutral C& ground state. The electrons move over the lattice sites through the transfer interaction between the nearestneighbor sites (T). The electrons and the lattice couple through the site-off-diagonal electron-lattice interaction constant (S). The model Hamiltonian is given as,
H=
- 1 T(E - tp’)(u&a,, + h-c.) (.c’.o - S C (Q, - Qp)(a,f,ar, L.I:a
+ h.c. - C&)
Here, I”(4 - /‘) is the transfer-energy of an electron between two sites / and L’ in the icosahedral symmetry of C,. a&(+,) is the creation (annihiIat~on) operator of an ekztron at site 8(r) with spin o(=a or /I ). S is the coup&g constant between the electron and the phonon localized at site d. &, is the coordinate of this phonon mode and P, is the momentum conjugate to &, while M denotes the effective mass for this mode and K is the harmonic spring constant with a phonon energy (mh@ which is given as w, = a. CL is the bond charge between nearest
Fig. I. Sketch of the @atom carbon cluster buckminsterfullerene structure. This structure has 60 vertices, 20 hexagonal faces, and 12 pentagonal faces. Each carbon atom occupies each vertex. There are two topoiogically different bond kngths: the length (r,) separating hexagons and the length frz) separating a hexagon from a pentagon.
Electron-phonon coupling on b~kminst~uli~ne
567
Table 1. Caiculated constants for the CC bonds in b~kminste~~ler~et Bond order, P, CC distance, II,., (A) From P,# From Coulson-Golebiewski selfconsistent Hiickei caIc. 1191 From ab ini& cak. [20] From expl. methods in aas _ 1211 _ in soIn. [21] in solid 1211 . -
0.64t(o.601)
0.455(0.479)
1.402(I .409)
1.435(1.431)
1.403 1.406
I.434 1.446
1.401 f 10 1.40 io.015 1.391 i 0.02
1.458 f 6 1.45 IO.OlS 1.455 -f 0.02
fThe values in parentheses are given when the electron hopping parameter (T) is chosen to the same irrespective to the difference of the bond distances. ~Coulson-Golebiewski’s jP” = 1.517 -O.lSOP,.
emnirical
formula
is
as
follows:
-
neighbor sites in the ground state for the neutral C,. Since we will consider only the ground states, we used the adiabatic approximation, neglecting the kinetic energy term of the lattice. In practical calculations, we assume that T(t - d’) between nearest neighbor sites in 30 bonds (ri) which lie solely in the six-membered ring @MR), is T, while that in 60 bonds (rr) which form the edges of both a five-membered ring (S-MR) and a 6-MR, is 0.9T (see Fig. 1). This set of transfer energies and CoulsonX+olebiewski’s empirical formula [18] ahnost reproduces the previous findings for the bond distance as shown in Table 1. Taking ho0 as the unit of energy, we transform all the quantities into the dimensionless form as
In these relations the as are appropriate labels for the N eigenve.ctors of h. Since the self-consistent values of (qfc > for ionic fullerenes are entirely different from those for neutral C, , graphite and diamond, our reformulation is based on a variational principle where both {qfJs) and {‘P,) are treated as variables. As for the extremum points {q,,,}, we apply the Feynma~~ellmann theorem subject to Z,,. q,,, = 0: since
h E H/ttw,,
with the Lagrange multiplier
q[ E d=jQ,
t(d - l’) = T(G - /‘)/ho,, and
s = S2&u~M.
t 3 T&o,, (2)
In terms of these dimensionless quantities, the new Hamiltonian h can be written as
(3) where m,, = a&u,,, , to, = t(8 - 6”) and qrr = q, - qp. Hereafter, we calculated the adiabatic potential c((qtl. f) of the lowest energy eigenstate, fG) of eqn (3) for eigenvalues of deformation {q,,,} and minimixed L with respect to the whole set (q,,,}. Denoting the eigenfunctions as l’y,> = X, !P;a: lvacuum) we apply the Schmidt o~hogonali~tion to obtain orthonormality and completeness relations for the follows: amplitudes as C, !PP;‘Y7. = a,, and ZI YJp;‘YpF = a,,, .
and
ac/aglc = 0 at &
C q,,, =
If
0,
we get &Y = fi
C (r+, ( 0
J = N-’
+ m$,, - C&) - J
c c (rntrs + m&A ZEDCC IJ’U
(4)
(3
Equations (3)-(S) can be solved by self-consistent iteration. As mentioned before, we are interested only in a general case of e-ph interaction in the ionic Cg(n = 1,2,3) ground states in which the value of s/t ranges from 0.1 to 2.0.
3. RESULTS AND DISCUSSION In our H~iltonian, we miss the explicit Coulomb interactions between x electrons. If the x-x Coulomb interactions are very strong, our approach is invalid and one should start from the “large-U” limit for the rr electrons. However, as shown in Table 1, the self-consistent solution for the C, neutral ground state led to the values rr (C,-C,) = 1.402 A for the bond length connecting 5-MRs (the bond fusing 6- and 6-MRs) and rr(C,-Cr) = 1.435 A for that within the S-MRs (i.e. for the bond fusing 5- and
KEE HAG
568
LEE and U-HYON PAEK
NL
(a)
C& grounddata
Fig. 2. Total energies, the number of localization (NL), and the number of bond charge difference (NBCD) inthe cY,- (n = 1,2,3) ground states. (a) Ck ground states; (b) C$ ground states; (c) C& ground states.
6-MRs). The result is in the range of the previous theoretical [19,20] and experimental [21] results. In the neutral ground state we also found that the icosahedral symmetry is preserved and the Fermi energy falls in the gap between a completely filled
five-fold
ci
CZ”
;
an empty
Cl
----
!
!
2
cso2-
and
!
1
!
State
state
___?_________-T-.__-.__.~ _.______ ---.._.___-
-----T-‘-
cso3-
degenerate
_-i_+--.i-~
I / D5d
CS
1 i
-----I7 Cso’-
ha I 0.4
i !I / i
threefold
degenerate state. In the ionic ground states for Cg (n = 1,2,3), the degenerate LUMOs of the neutral C, ground state are partially filled. As is well known this kind of
c2v I
1 cs / I i
0.8
i i i
I ) !
c2v
1 I
I
1.2
1.6
i
c2
I / f I_
2.0
s/t Fig. 3. Deformation lattices as a function of s/t values in the C& (n = 1,2,3) ground states.
Electron-phonon coupling on buckminsterfullerene ionic system relaxes to the lattice configuration with reduced symmetry. For a given number of electrons, the self-consistent solution is obtained from eqn (3), which has 20 different s/t values from 0.1 to 2.0. Total energies, number of localization (NL), and number of bond charge difference (NBCD) for C& (n = 1,2,3) ground states are shown in Fig. 2. Here, NL defined as
NL =
where L is a site, S,( = l&12) is a localized charge density and S, (= &12) is the same charge density with electron-phonon coupling constant being zero, was calculated for each unique s/t. Here MOs were orthonomalixed by the Schmidt method. NBCD defined as NBCD =
{~WJ’ld+ll)d~} where
(a) N&ml
-
{(BC), - (BCC,)k}2,
(7)
BC is the bond charge between nearest i-site and j-site in the C$(n = 1,2,3) ground states and BCC&, is the same bond charge in the neutral C, ground states, was calculated for each unique s/t. As shown in Fig. 2, the self-consistent solution in this case is that there are three regions based on electron-phonon coupling strength, with small, intermediate and large values of the electron-phonon coupling constant, respectively. From the self-consistent calculation, we obtain diverse symmetries of deformed lattices for the C!! (n = 1,2,3) ground states as a function of s/t and show their features in Fig. 3. The energy levels and their degeneracies for each case of relaxed geometries in neutral and ionic ground states are shown in Fig. 4. In the region of s/t values of 0.5 and downward in the ionic C, and Cp ground states, and of 0.4 and downward in the ionic C& ground states, the electron-phonon coupling is a perturbation to the electron-hopping and elastic energy terms, which is the physical origin of the Jahn-Teller effect.
0.0
Ca ground 8tata
Fig. 4(a).
0.3
1 (i,,) = k
(6) neighbor
s/t
569
0.4
od
0.4
0.7
(b) C&,,‘- ground statea Fig. 4(b).-continued on next page
KEE HAG LEE and U-HYON
(o) &i
ground
PAEK
&ten
Fig. 4(c).
,(d) &
ground
st&a
Fig. 4(d). Fig. 4. Energy levels and the degeneracies (measured by the length of each parallel bar of the eigenstates with, eigenenergies fEi/t) in relaxed geometries related to specifk s/r values for the cases of: (a) neutral ground states; (b) Cb ground states; (c) C& ground states; and (d) C;b- ground states.
In the above regions in all three systems, total energies are linearly dependent upon the s/t values, but the slopes are very small. In the triple degenerate states, the splitting of the triplet state agrees with the analytical solution obtained from the application of the first order perturbation theory to degenerate cases. That the I,, levels split, for example, as indicated in Fig. 4 agrees with the analytical solution by applying the first order
perturbation
to the t,, degenerate states, that is
1~~
%Z2
.~,l_o
where Q is the energy for I,, degenerate state and I’,, is the perturbed term based on electron-phonon coupling. For Vii= V < 0 (i,j = I, 2,3) in C, and
Electron-phonon
coupling on bu~k~nsterfuil~rene
571
KEE HAG Las and U-HYON PAEK
512
C& ground states, the r,, levels split the qu and e,, levels and the corresponding splitting energies are -2V and V for uru and e,, levels, respectively. As shown in Fig. 4, this results in a distortion which has DSd symmetry, picking out a radial direction through the center of a pentagon. The splitting of the t,, level by the electron-phonon coupling in these regions of C, and Cg matches the result of Varma et al. [ll] and Schluter ef al. [l l] which is based on the Jahn-Teller effect. Meanwhile, in the case of C& ground states, e,, levels split further two a, levels resulting in a distortion which has Ci symmetry, but the distorted lattices are very similar to the shape of D,, symmetry which is the result in C, and C& ground states. Here the splitting upholds the claim of Schluter et al. [ 111, that the third electron doped to of the C, not only changes the magnitude Jahn-Teller distortion but also changes its form. Thus, in the third electron doped case, our result is different from that of Vat-ma et al. [1 11. But in the regions of intermediate and strong electron-phonon coupling, each Vijdoes not have the same value. Thus each degenerate level splits nondegenerate levels. Also, this gives rise to the deformed lattices of localized self-trapped electronic states. In the region where the electron-phonon coupling strength is strong, the total energies for all three systems are linearly and largely dependent upon the s/t values. How the localization of C&(n = 1,2,3) ground states varies with the electron-phonon coupling strength is shown in Fig. 2. The total patterns for NL in three ionic Cso ground state are approximately the same. The s/t value which has a maximum NL is 1.2 for C,, 1.1 for C& and 1.0 for C& ground states. When we pass the maximum NL value the state becomes more delocalized. The larger s/t values have the larger NBCD in the C& (n = 1,2,3) ground states. The shapes of NBCD are approximately the same for these three systems. The s/t values which have the maximum NL values in the C& (n = 1,2,3) ground states have the maximum variation between nearest neighbor NBCDs. After we pass these specific values, the NBCD is almost unchanged.
As shown in Fig. 4, there are 20 panels for 20 different values of s/t from 0.1 to 2.0. In each panel the position of each parallel bar marks the value of one eigenenergy and the height of the bar measures the corresponding degeneracy. The widths of two eigenenergies at the ends of each spectrum as a function of s/t in the C!& (n = 1,2,3) ground states are divided approximately into three regions as follows: an approximately constant region up to 0.5 for s/t; a small decreased region with 0.6-0.9 for s/t; and a large increased region with 1.0-2.0 for s/t. charge distribution on C& The electron (n = 1,2,3) ground states as shown in Fig. 5 is used to represent deformed lattices. For illustrative purpose here, we use a Schlegel diagram of the three-dimensional system. In C; and C& ground states with s/t values up to 0.5, the deformed lattices (Fig. 5) have D,, symmetry, (but not among the crystallographic groups) which has a fivefold rotation axis, five twofold rotation axes and inversion symmetry elements. The four- and fivefold degenerate states, which are in the vector representation of the icosahedral group (I,,), split into 2 + 2 and 2 + 2 + 1, respectively, as the threefold degenerate states split into onefold and twofold states. Here, the T,, state for the LUMOs splits into A,, and E,, states, and the Z-Z,state for the HOMOs into A,,, E,, and E,, states as shown in Fig. 4. In a range of 0 < s/t < 0.4 in C& ground states, the deformed lattices are of Ci symmetry which has only the inversion symmetry element as shown in Fig. 5. The splitting of energy degeneracies in I,, symmetry during lattice distortion is related to the symmetry species of icosahedral subgroups D, , C,, , C, , C,, C,, and C, as shown in Table 2. Table 2 also shows the splitting levels and represents the symmetry of energy levels as shown in Fig. 4. Mele et al. [13] have shown that C, has a small electron-phonon coupling constant with C,, symmetry in which the threefold degenerate states split into onefold and twofold states with inversion symmetry element. Here we see that their deformed lattice may be different from C,, symmetry because C,, symmetry has neither a two-
Table 2. Decomposition of I,, relative to its subgroups D,, C,,, C,, C,, Ci and C, Ih
D,
4
TIE 7-21 G8 H8 A” T,” 7’2,
G” H”
C2” A,
Al8
AQ + E,, A, + E2, EI, + Ezg 4,+4,+E2p
Al” A,, + E,, 4,
+
~42
E2u
En, + E2u AI, + E,, +
B,+B,+A, B,+B2+A2 A,+A,+B,+B2 2A,,+A,+B,+B2
E2u
A,+B,+B* A,+B,+B, A,+A,+B,+B2 A,+2A2+B,+B2
C2
A A +2B A +2B 2A+2B 3A+2B A A +2B A +2B 2A+2B 3A-k28
G
A’ A’+2A” A’+ 2A” 2A‘+ 2A” 3A’+2A” A” ZA’+A” 2A’+A” 2A’+2A” 2A’+3A”
C,
Cl
A, 3A, 3A, 4A, 5A, Ai 3A, 3A, 4A, SA,
A 3A 3A 4A 5A A 3A 3A 4A SA
Electron-phonon
coupling on buckminsterfullerene
fold degeneracy state nor an inversion symmetry element. From our results, shown in Table 2 and Fig. 4, we suggest that the deformed lattices have L), symmetry. It is interesting to note that the same electron-phonon coupling strength, which has the same s/t value, does not give the same deformed lattices in the Cn& (n = 1,2,3) ground states. Even in the region of weak electron-phonon coupling, the C& compared to the C; and C& ground states is differently deformed. In conclusion, the Cg (n = 1,2,3) ground states undergo Jahn-Teller distortions in the weak electron-phonon coupling region, while lower-energy self-localized electronic states occur in the intermediate and strong electron-phonon coupling regions. Acknowfedgemenrs-We are very grateful to Prof. K. Nasu For his hospitality, encouragement and valuable discussions during a stay in his laboratory at the Institute for Molecular Science, Okazaki, Japan. This work is supported by the Japan Society for the Promotion of science, Korea Seienoe and Engineering Foundation and the Joint Studies Program of IMS. The authors thank the Computer Center, IMS for the use of the HITAC M-680H and S-820/80 computers.
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1993
oDz2-3#7/93 sf&i + 0.00 g’: 1993 Pergamon Press ttd
ELECTRON-PHONON COUPLING ON THE IONIC BO-ATOM CARBON CLUSTER BUCKMINSTERFULLERENE U-HYON PAEK$
KEE HAG LE@ and
@Iq3artrnent of Chemistry, ~o~wang
University, Iri, 570-749, Kwea ffkpartment of chemistry, Gyeongsang Nationaf University, Jinju 660-701, Korea (Receiued I May I992; accepted in revised j&m 12 November 1992) Abafract-Using a tight-binding model with electron-phonon coupling based on the modified SSH model Hamiltonian, we studied several ground states of the neutral and ionic 60-atom carbon cluster, buc~nsterfull~rene. The ground states of ionic structures are shown to consist of Jahn-Teller distortion and localized self-trapped electron states dependent upon the ratio of the electron-phonon coupling constant to the electron hopping constant. Keywords:
Ionic C,
ground state, tight-binding methods, electron-phonon
coupling, Jahn-Teller distor-
tion, self-localizat&
The recent synthesis of macroscopic quantities [I] of the C, fullerenes has made the discovery of the chemical and physical properties of this new molecular form of carbon possible. Spectroscopic [Z] and diffraction ]3] studies have confirmed the predicted truncated icosahedrai structure [4] of the C, molecule. Recently, the formation of new semiconducting (51, high conducting ]6] and superconducting [?j anionic charge transfer compounds of C, witb molecular and a&ah metal counterions, respeetive]y, has been discovered. Hofzer er trf &rj have found that the approximate composition of M3Cso (M = K, Rb) is the superconducting phase, and neither under- nor over-doped phases are superconducting. Phot~mission studies of K,C, emphasize changes in the valence bands, correctly noting the filling of the lowest unoccupied molecular orbitals (LUMOs) derived bands [8], and estimate their stoichiometries from the emission intensities of band features derived from the highest occupied molectdar orbitals (HOMOs) and the LWMOs. Renning et & [g] have suggested that this filling is not rigid band-hke and K,C, is a strong electron-phonon coupling superconductor. Poirier et al. [S] and Chen ef af. [9] reported that K,Cso samples with intermediate stoichiometries represented two-phase regions of an ~~l~b~urn phase diagrams. Chen et al. [9] also suggested that these systems are weakly coupled BCS-like superconductors. With synchrotron X-ray powder diffraction, McCauley ‘et al. [lO] reported that the miscibility gap in I&C, implied by photoemission results [9] did not cover the entire Otx<3 range.
Tanigaki et al. [q and Fleming e? al. [7j reported that the variation of ?Y,with dopant supports the interpretation that the transition temperature of these fullerides is determined mainly by the density of states at the Fermi level. Duclos et al. [I I] reported that the Raman fr~uen~es for doped M,C, (where M is K, Rb or Cs) films are very similar, indicating little change in intramolecular phonon modes. Irrespective of the details of the precise microscopic mechanism of superconductivity in alkali metal doped C,, the density of states at the Fermi level and electron--phonon interactions would play a crucial role in determining their high T,-values 17, 1I]. ft is therefore reasonable to assume that this would be consistent with the idea that C, intramolecular phonon modes in a single molecule are a very important factor of su~~onducti~ty in the doped Cso systems. This has already been suggested by !&hluter et al., Varma et al. and Anderson et al. [ 111.They find that tangential atomic vibrations (with C atoms moving on the sphere surface) coupled more strongly with electrons. In intramolecular phonon theories of ~~r~nducti~ty in I&C,, Schfuter ef ai. fl 1) and Varma et uf. [l l] have shown that electron scattering is dominated by particular on-ball Jahn-Teller-type modes on the scale of the large on-ball n-hopping energy. Here Varma et al. [l l] calculate the characteristic intramol~ular defo~ation potential per unit displacement of the specific mode which is based on the Jahn-Teller deformations of a C, molecule. Schluter ef al. [ 1l] shows that the third electron not only changes tbe magnitude of the Jahn-Teller distortion but also changes its form. Therefore, it is interesting to see the Jahn-Teller distortion potential
565
566
KEEHAG LEE and U-HYMNPAEK
of C& (where n = I, 2,3) as a function of the doping concentration. Wehner and Van Zee f12] and Kroto et al. 1123 have presented a comprehensive overview of the theoretical work on fullerene-60. Hayden and Mele [13] have studied several ground and excited states of the neutral and ionic C,, cluster using the tightbinding mode1 with electron-phonon coupling. They found that the singly-ionic ground state underwent Jahn-Teller distortion, but that no localized selftrapped electronic states occurred. Negri et al. [14] have classified Jahn-Teller instabilities in both excited neutral and ground ionic electronic states. Recentiy, Clementi et al. fi5] performed ab i&if3 calculations for the ground states of the neutral and ionic C, cluster within the constraint of ih geometry. As far as we are aware, the general effect of electron-phonon coupling in ionic Cm ground states is not known. It would be interesting to see if electrons in ionized ground states of reduced symmetry have bcalized or extended character in a system intermediate in size between small clusters and solids. The soliton model of Su-Shriffer-Heeger (SSH), which is the tight-binding mode1 with electron-phonon coupling, demonstrates the novel phenomena with the midgap state in ~1ya~tyIene f16]. In this paper, we present the results from self-consistent numerical calculations which allow for a complete relaxation of all n: electrons and individual atoms in the ground state of neutral and ionized C, systems. Our self-consistent, orthogonahzed linear ~rn~nati~n of atomic orbitals (0LCAO) method is based on the SSH Hamiltonian model, which hereafter is called the modified SSH model Hamiltonian. The variation of lattice deformation in each system is considered as a function of the ratio of the electron-phonon coupling constant (S) to the ~~~~on-hopping constant (T). The iocalized or extended character of the ekectronic states with the lattice deformation is also discussed. We construct the model Hamiltonian calculation scheme in section 2. The self-consistent computation, which is similar to the method proposed by Shastry flY& and the Schmidt o~hogona~~tion of alI 7corbitals in order to consider only the lattice vibration modes are used in our approach. The particular utility of our model calculations lies in the speed and ease with which they yield correlation diagrams. The criterion for convergermy was set to JO-‘* for the charge iteration to ensure a highly refined structure for all calculations. But for C& ionic ground states only in the range of 0.1-0.4 for s/t values, we obtained the convergency with lo-’ for the charge iteration. Ait our results will be compared to the findings of other authors.
2. MODEL HAMILTONIAN CALCULATION SCWEME As a model for neutral and ionized C, systems, we consider a buckminsterfullerene composed of 60 lattice sites and N electrons (N 3 60). Each site has one electronic state per spin direction corresponding to the carbon x orbital of OLCAO in the neutral C& ground state. The electrons move over the lattice sites through the transfer interaction between the nearestneighbor sites (T). The electrons and the lattice couple through the site-off-diagonal electron-lattice interaction constant (S). The model Hamiltonian is given as,
H=
- 1 T(E - tp’)(u&a,, + h-c.) (.c’.o - S C (Q, - Qp)(a,f,ar, L.I:a
+ h.c. - C&)
Here, I”(4 - /‘) is the transfer-energy of an electron between two sites / and L’ in the icosahedral symmetry of C,. a&(+,) is the creation (annihiIat~on) operator of an ekztron at site 8(r) with spin o(=a or /I ). S is the coup&g constant between the electron and the phonon localized at site d. &, is the coordinate of this phonon mode and P, is the momentum conjugate to &, while M denotes the effective mass for this mode and K is the harmonic spring constant with a phonon energy (mh@ which is given as w, = a. CL is the bond charge between nearest
Fig. I. Sketch of the @atom carbon cluster buckminsterfullerene structure. This structure has 60 vertices, 20 hexagonal faces, and 12 pentagonal faces. Each carbon atom occupies each vertex. There are two topoiogically different bond kngths: the length (r,) separating hexagons and the length frz) separating a hexagon from a pentagon.
Electron-phonon coupling on b~kminst~uli~ne
567
Table 1. Caiculated constants for the CC bonds in b~kminste~~ler~et Bond order, P, CC distance, II,., (A) From P,# From Coulson-Golebiewski selfconsistent Hiickei caIc. 1191 From ab ini& cak. [20] From expl. methods in aas _ 1211 _ in soIn. [21] in solid 1211 . -
0.64t(o.601)
0.455(0.479)
1.402(I .409)
1.435(1.431)
1.403 1.406
I.434 1.446
1.401 f 10 1.40 io.015 1.391 i 0.02
1.458 f 6 1.45 IO.OlS 1.455 -f 0.02
fThe values in parentheses are given when the electron hopping parameter (T) is chosen to the same irrespective to the difference of the bond distances. ~Coulson-Golebiewski’s jP” = 1.517 -O.lSOP,.
emnirical
formula
is
as
follows:
-
neighbor sites in the ground state for the neutral C,. Since we will consider only the ground states, we used the adiabatic approximation, neglecting the kinetic energy term of the lattice. In practical calculations, we assume that T(t - d’) between nearest neighbor sites in 30 bonds (ri) which lie solely in the six-membered ring @MR), is T, while that in 60 bonds (rr) which form the edges of both a five-membered ring (S-MR) and a 6-MR, is 0.9T (see Fig. 1). This set of transfer energies and CoulsonX+olebiewski’s empirical formula [18] ahnost reproduces the previous findings for the bond distance as shown in Table 1. Taking ho0 as the unit of energy, we transform all the quantities into the dimensionless form as
In these relations the as are appropriate labels for the N eigenve.ctors of h. Since the self-consistent values of (qfc > for ionic fullerenes are entirely different from those for neutral C, , graphite and diamond, our reformulation is based on a variational principle where both {qfJs) and {‘P,) are treated as variables. As for the extremum points {q,,,}, we apply the Feynma~~ellmann theorem subject to Z,,. q,,, = 0: since
h E H/ttw,,
with the Lagrange multiplier
q[ E d=jQ,
t(d - l’) = T(G - /‘)/ho,, and
s = S2&u~M.
t 3 T&o,, (2)
In terms of these dimensionless quantities, the new Hamiltonian h can be written as
(3) where m,, = a&u,,, , to, = t(8 - 6”) and qrr = q, - qp. Hereafter, we calculated the adiabatic potential c((qtl. f) of the lowest energy eigenstate, fG) of eqn (3) for eigenvalues of deformation {q,,,} and minimixed L with respect to the whole set (q,,,}. Denoting the eigenfunctions as l’y,> = X, !P;a: lvacuum) we apply the Schmidt o~hogonali~tion to obtain orthonormality and completeness relations for the follows: amplitudes as C, !PP;‘Y7. = a,, and ZI YJp;‘YpF = a,,, .
and
ac/aglc = 0 at &
C q,,, =
If
0,
we get &Y = fi
C (r+, ( 0
J = N-’
+ m$,, - C&) - J
c c (rntrs + m&A ZEDCC IJ’U
(4)
(3
Equations (3)-(S) can be solved by self-consistent iteration. As mentioned before, we are interested only in a general case of e-ph interaction in the ionic Cg(n = 1,2,3) ground states in which the value of s/t ranges from 0.1 to 2.0.
3. RESULTS AND DISCUSSION In our H~iltonian, we miss the explicit Coulomb interactions between x electrons. If the x-x Coulomb interactions are very strong, our approach is invalid and one should start from the “large-U” limit for the rr electrons. However, as shown in Table 1, the self-consistent solution for the C, neutral ground state led to the values rr (C,-C,) = 1.402 A for the bond length connecting 5-MRs (the bond fusing 6- and 6-MRs) and rr(C,-Cr) = 1.435 A for that within the S-MRs (i.e. for the bond fusing 5- and
KEE HAG
568
LEE and U-HYON PAEK
NL
(a)
C& grounddata
Fig. 2. Total energies, the number of localization (NL), and the number of bond charge difference (NBCD) inthe cY,- (n = 1,2,3) ground states. (a) Ck ground states; (b) C$ ground states; (c) C& ground states.
6-MRs). The result is in the range of the previous theoretical [19,20] and experimental [21] results. In the neutral ground state we also found that the icosahedral symmetry is preserved and the Fermi energy falls in the gap between a completely filled
five-fold
ci
CZ”
;
an empty
Cl
----
!
!
2
cso2-
and
!
1
!
State
state
___?_________-T-.__-.__.~ _.______ ---.._.___-
-----T-‘-
cso3-
degenerate
_-i_+--.i-~
I / D5d
CS
1 i
-----I7 Cso’-
ha I 0.4
i !I / i
threefold
degenerate state. In the ionic ground states for Cg (n = 1,2,3), the degenerate LUMOs of the neutral C, ground state are partially filled. As is well known this kind of
c2v I
1 cs / I i
0.8
i i i
I ) !
c2v
1 I
I
1.2
1.6
i
c2
I / f I_
2.0
s/t Fig. 3. Deformation lattices as a function of s/t values in the C& (n = 1,2,3) ground states.
Electron-phonon coupling on buckminsterfullerene ionic system relaxes to the lattice configuration with reduced symmetry. For a given number of electrons, the self-consistent solution is obtained from eqn (3), which has 20 different s/t values from 0.1 to 2.0. Total energies, number of localization (NL), and number of bond charge difference (NBCD) for C& (n = 1,2,3) ground states are shown in Fig. 2. Here, NL defined as
NL =
where L is a site, S,( = l&12) is a localized charge density and S, (= &12) is the same charge density with electron-phonon coupling constant being zero, was calculated for each unique s/t. Here MOs were orthonomalixed by the Schmidt method. NBCD defined as NBCD =
{~WJ’ld+ll)d~} where
(a) N&ml
-
{(BC), - (BCC,)k}2,
(7)
BC is the bond charge between nearest i-site and j-site in the C$(n = 1,2,3) ground states and BCC&, is the same bond charge in the neutral C, ground states, was calculated for each unique s/t. As shown in Fig. 2, the self-consistent solution in this case is that there are three regions based on electron-phonon coupling strength, with small, intermediate and large values of the electron-phonon coupling constant, respectively. From the self-consistent calculation, we obtain diverse symmetries of deformed lattices for the C!! (n = 1,2,3) ground states as a function of s/t and show their features in Fig. 3. The energy levels and their degeneracies for each case of relaxed geometries in neutral and ionic ground states are shown in Fig. 4. In the region of s/t values of 0.5 and downward in the ionic C, and Cp ground states, and of 0.4 and downward in the ionic C& ground states, the electron-phonon coupling is a perturbation to the electron-hopping and elastic energy terms, which is the physical origin of the Jahn-Teller effect.
0.0
Ca ground 8tata
Fig. 4(a).
0.3
1 (i,,) = k
(6) neighbor
s/t
569
0.4
od
0.4
0.7
(b) C&,,‘- ground statea Fig. 4(b).-continued on next page
KEE HAG LEE and U-HYON
(o) &i
ground
PAEK
&ten
Fig. 4(c).
,(d) &
ground
st&a
Fig. 4(d). Fig. 4. Energy levels and the degeneracies (measured by the length of each parallel bar of the eigenstates with, eigenenergies fEi/t) in relaxed geometries related to specifk s/r values for the cases of: (a) neutral ground states; (b) Cb ground states; (c) C& ground states; and (d) C;b- ground states.
In the above regions in all three systems, total energies are linearly dependent upon the s/t values, but the slopes are very small. In the triple degenerate states, the splitting of the triplet state agrees with the analytical solution obtained from the application of the first order perturbation theory to degenerate cases. That the I,, levels split, for example, as indicated in Fig. 4 agrees with the analytical solution by applying the first order
perturbation
to the t,, degenerate states, that is
1~~
%Z2
.~,l_o
where Q is the energy for I,, degenerate state and I’,, is the perturbed term based on electron-phonon coupling. For Vii= V < 0 (i,j = I, 2,3) in C, and
Electron-phonon
coupling on bu~k~nsterfuil~rene
571
KEE HAG Las and U-HYON PAEK
512
C& ground states, the r,, levels split the qu and e,, levels and the corresponding splitting energies are -2V and V for uru and e,, levels, respectively. As shown in Fig. 4, this results in a distortion which has DSd symmetry, picking out a radial direction through the center of a pentagon. The splitting of the t,, level by the electron-phonon coupling in these regions of C, and Cg matches the result of Varma et al. [ll] and Schluter ef al. [l l] which is based on the Jahn-Teller effect. Meanwhile, in the case of C& ground states, e,, levels split further two a, levels resulting in a distortion which has Ci symmetry, but the distorted lattices are very similar to the shape of D,, symmetry which is the result in C, and C& ground states. Here the splitting upholds the claim of Schluter et al. [ 111, that the third electron doped to of the C, not only changes the magnitude Jahn-Teller distortion but also changes its form. Thus, in the third electron doped case, our result is different from that of Vat-ma et al. [1 11. But in the regions of intermediate and strong electron-phonon coupling, each Vijdoes not have the same value. Thus each degenerate level splits nondegenerate levels. Also, this gives rise to the deformed lattices of localized self-trapped electronic states. In the region where the electron-phonon coupling strength is strong, the total energies for all three systems are linearly and largely dependent upon the s/t values. How the localization of C&(n = 1,2,3) ground states varies with the electron-phonon coupling strength is shown in Fig. 2. The total patterns for NL in three ionic Cso ground state are approximately the same. The s/t value which has a maximum NL is 1.2 for C,, 1.1 for C& and 1.0 for C& ground states. When we pass the maximum NL value the state becomes more delocalized. The larger s/t values have the larger NBCD in the C& (n = 1,2,3) ground states. The shapes of NBCD are approximately the same for these three systems. The s/t values which have the maximum NL values in the C& (n = 1,2,3) ground states have the maximum variation between nearest neighbor NBCDs. After we pass these specific values, the NBCD is almost unchanged.
As shown in Fig. 4, there are 20 panels for 20 different values of s/t from 0.1 to 2.0. In each panel the position of each parallel bar marks the value of one eigenenergy and the height of the bar measures the corresponding degeneracy. The widths of two eigenenergies at the ends of each spectrum as a function of s/t in the C!& (n = 1,2,3) ground states are divided approximately into three regions as follows: an approximately constant region up to 0.5 for s/t; a small decreased region with 0.6-0.9 for s/t; and a large increased region with 1.0-2.0 for s/t. charge distribution on C& The electron (n = 1,2,3) ground states as shown in Fig. 5 is used to represent deformed lattices. For illustrative purpose here, we use a Schlegel diagram of the three-dimensional system. In C; and C& ground states with s/t values up to 0.5, the deformed lattices (Fig. 5) have D,, symmetry, (but not among the crystallographic groups) which has a fivefold rotation axis, five twofold rotation axes and inversion symmetry elements. The four- and fivefold degenerate states, which are in the vector representation of the icosahedral group (I,,), split into 2 + 2 and 2 + 2 + 1, respectively, as the threefold degenerate states split into onefold and twofold states. Here, the T,, state for the LUMOs splits into A,, and E,, states, and the Z-Z,state for the HOMOs into A,,, E,, and E,, states as shown in Fig. 4. In a range of 0 < s/t < 0.4 in C& ground states, the deformed lattices are of Ci symmetry which has only the inversion symmetry element as shown in Fig. 5. The splitting of energy degeneracies in I,, symmetry during lattice distortion is related to the symmetry species of icosahedral subgroups D, , C,, , C, , C,, C,, and C, as shown in Table 2. Table 2 also shows the splitting levels and represents the symmetry of energy levels as shown in Fig. 4. Mele et al. [13] have shown that C, has a small electron-phonon coupling constant with C,, symmetry in which the threefold degenerate states split into onefold and twofold states with inversion symmetry element. Here we see that their deformed lattice may be different from C,, symmetry because C,, symmetry has neither a two-
Table 2. Decomposition of I,, relative to its subgroups D,, C,,, C,, C,, Ci and C, Ih
D,
4
TIE 7-21 G8 H8 A” T,” 7’2,
G” H”
C2” A,
Al8
AQ + E,, A, + E2, EI, + Ezg 4,+4,+E2p
Al” A,, + E,, 4,
+
~42
E2u
En, + E2u AI, + E,, +
B,+B,+A, B,+B2+A2 A,+A,+B,+B2 2A,,+A,+B,+B2
E2u
A,+B,+B* A,+B,+B, A,+A,+B,+B2 A,+2A2+B,+B2
C2
A A +2B A +2B 2A+2B 3A+2B A A +2B A +2B 2A+2B 3A-k28
G
A’ A’+2A” A’+ 2A” 2A‘+ 2A” 3A’+2A” A” ZA’+A” 2A’+A” 2A’+2A” 2A’+3A”
C,
Cl
A, 3A, 3A, 4A, 5A, Ai 3A, 3A, 4A, SA,
A 3A 3A 4A 5A A 3A 3A 4A SA
Electron-phonon
coupling on buckminsterfullerene
fold degeneracy state nor an inversion symmetry element. From our results, shown in Table 2 and Fig. 4, we suggest that the deformed lattices have L), symmetry. It is interesting to note that the same electron-phonon coupling strength, which has the same s/t value, does not give the same deformed lattices in the Cn& (n = 1,2,3) ground states. Even in the region of weak electron-phonon coupling, the C& compared to the C; and C& ground states is differently deformed. In conclusion, the Cg (n = 1,2,3) ground states undergo Jahn-Teller distortions in the weak electron-phonon coupling region, while lower-energy self-localized electronic states occur in the intermediate and strong electron-phonon coupling regions. Acknowfedgemenrs-We are very grateful to Prof. K. Nasu For his hospitality, encouragement and valuable discussions during a stay in his laboratory at the Institute for Molecular Science, Okazaki, Japan. This work is supported by the Japan Society for the Promotion of science, Korea Seienoe and Engineering Foundation and the Joint Studies Program of IMS. The authors thank the Computer Center, IMS for the use of the HITAC M-680H and S-820/80 computers.
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