Group theoretical prediction
Before the calculation of band structures, the splitting or degeneracy of energy band is predicted from a group theory examination.
Figure 3 illustrates the B.Z. of
Rh-C
60 crystal. The symmetry of this B.Z. is
D3d, which is a subgroup of
Ih.
Figure 3.
The reciprocal space around the first B.Z. The cross points (×) denote reciprocal lattice points.
Figure 3.
The reciprocal space around the first B.Z. The cross points (×) denote reciprocal lattice points.
Under the empty lattice approximation, the Bloch function is expressed as
for which the energy corresponding to this wavefunction is
where
k and
K denote wave number and reciprocal lattice vectors respectively. When the behavior at Γ-point is in the question,
k set to be zero in eq.(2). Then the following eight functions (3a-h) which satisfy
E =
E0 were constructed,
By operating the elements that belonged to the group of Γ in
Table 2, the following reducible representation was obtained and it could be resolved as
Table 2.
Character table for G-point of rhombohedral lattice
Table 2.
Character table for G-point of rhombohedral lattice
Γ | E | 2C3 | 3C2 | i | 2S6 | 3σd |
---|
Γ1 | 1 | 1 | 1 | 1 | 1 | 1 |
Γ2 | 1 | 1 | -1 | 1 | 1 | -1 |
Γ3 | 2 | -1 | 0 | 2 | -1 | 0 |
Γ4 | 1 | 1 | 1 | -1 | -1 | -1 |
Γ5 | 1 | 1 | -1 | -1 | -1 | 1 |
Γ6 | 2 | -1 | 0 | -2 | 1 | 0 |
Using the projection operator method[
18], the following linear combination of Bloch functions were obtained.
Using the following relations, cos
x ≈ 1+
x and sin
x ≈
x, these wavefunctions can be rewritten as
According to the character table of Ih which pristine C60 belongs to, the three-fold degenerate LUMOs split into a singly-degenerate ѱ(Γ5) and two-fold degenerate ѱ(Γ6), whilst the five-fold degenerate HOMOs split into a singly-degenerate ѱ(Γ4) and two-fold degenerate ѱ(Γ1) and ѱ(Γ3). These predictions were confirmed by the following band calculations.
Rh-C60 without distortion of C60 cage
Figure 4 illustrates the band structure of normal
Rh-C
60 i.e. without cage distortion. Each special points were defined as Γ (0, 0, 0), M (1/2, 0, 0), K (1/3, 1/3, 0), A (0, 0, 1/2) as illustrated in the B.Z. Valence and conduction bands were separated by the broken line ( E
f = Fermi level).
Figure 4.
(A) The band structure of normal Rh-C60 (d = 0%); (B) The first B.Z. with special points and line. The detail information is given in the text.
Figure 4.
(A) The band structure of normal Rh-C60 (d = 0%); (B) The first B.Z. with special points and line. The detail information is given in the text.
The two lowest conduction bands (LCB) were essentially two-fold degenerate at the Γ point. According to the IR-active modes[
6], we confirmed that LCB showed the interaction of C*-C* bond between C
60 units. Conversely, the highest valence band (HVB) was non-degenerate. This split of HVB from the five-fold degenerated
Hg came from anisotropic structure of the rhombohedral phase. Namely, LCB and HVB corresponded to (x,y) and z
2 symmetry respectively. Therefore if the LCB are stabilized, they can get closer to the HVB without repulsion because they are mutually orthogonal. The calculated band gap was 1.23 eV, reproducing the semi-conducting nature of normal
Rh-C
60.
Rh-C60 with distortion of C60 cage
Since the LCB expresses the inter-fullerene interaction, it is expected that these LCB will be stabilized when the distance of inter-fullerene become shorter. A distortion parameter
d was defined as
where D and D
0 denote the inter-fullerene distance with and without distortion, respectively (i.e. D
0 = 2.408Å, D < 2.408Å). By varying the parameter
d, the LCB were stabilized as shown in
Figure 5.
Table 3 shows the variation in the Fermi energy (E
f) and band gap (E
g) as a function of this distortion parameter.
Figure 5.
The band structures of Rh-C60 with distortion (A) d = 2.08% (B) d = 4.15% (C) d = 8.30% (D) The Fermi-Dirac distribution under the distrtion d = 8.30%. The metallic character was expected judging from (C) and (D).
Figure 5.
The band structures of Rh-C60 with distortion (A) d = 2.08% (B) d = 4.15% (C) d = 8.30% (D) The Fermi-Dirac distribution under the distrtion d = 8.30%. The metallic character was expected judging from (C) and (D).
Table 3.
Calculated band gaps* (Eg) and Fermi level* (EF) with variation of distortion parameter d
Table 3.
Calculated band gaps* (Eg) and Fermi level* (EF) with variation of distortion parameter d
d | Eg | EF |
---|
0.00 | 1.23 | -2.30 |
2.08 | 1.00 | -2.30 |
4.15 | 0.75 | -2.30 |
8.30 | 0.31 | -2.24 |
Figure 6.
The upper four band structures (X-s: X=A, B, C, D) correspond singlet Rh-C60 with distortion (A) d = 9.14, (B) d = 9.97, (C) d = 10.80, (D) d = 11.63, respectively. Similarly, the lower four band structures (X-t: X=A, B, C, D) correspond triplet Rh-C60 with distortion (A) d = 9.14, (B) d = 9.97, (C) d = 10.80, (D) d = 11.63, respectively. Atomic unit was used for energy. The band structure (D-s) includes flat band.
Figure 6.
The upper four band structures (X-s: X=A, B, C, D) correspond singlet Rh-C60 with distortion (A) d = 9.14, (B) d = 9.97, (C) d = 10.80, (D) d = 11.63, respectively. Similarly, the lower four band structures (X-t: X=A, B, C, D) correspond triplet Rh-C60 with distortion (A) d = 9.14, (B) d = 9.97, (C) d = 10.80, (D) d = 11.63, respectively. Atomic unit was used for energy. The band structure (D-s) includes flat band.
An analysis of these results indicates that the greater distortion of the C
60 cage leads to a decrease in the band gap, E
g. Considering the Fermi-Dirac distribution at room temperature (T=300K),
Rh-C
60 is expected to become metallic in agreement with the experimentally observed semiconductor-metal phase transition [
8]. It is noteworthy that the dispersion of the HVB is considerably small, that is, a flat band was observed. The LCB and this HVB became closer and closer as
d increased. Of particular note is the case when
d =11.6%. At this point one of the LCB and the HVB form a two-fold degenerate half-filled flat bands around the Fermi level as shown in
Figure 6.
According to the Mielke-Tasaki theorem [
17], this band structure satisfies the necessary condition for flat band ferromagnetism. Thus if on-site Coulomb repulsion
U on fullerene is not zero, the ferromagnetic electronic state may be one of the possible ground states. In order to elucidate the possibility of ferromagnetic ground state, an examination of the relative stability of this ferromagnetic state is necessary.
Figure 7 shows the relative energies of the singlet (diamagnetic) and triplet (ferromagnetic) states per unit cell. From
Figure 7, we observed four distinct regions as follows: (1) For small
d (0 ≤
d ≤ 9.5 %, labelled A in
Figure 7), the singlet state was more stable than triplet state. Since normal
Rh-C
60 was considered to be semi-conducting, this is consistent with experimental results. (2) An energy minimum for the triplet state at
d = 2%. The energy of this triplet is stabilized by 0.06 eV compared to the undistorted structure (
d=0%). However the triplet is still substantially higher in energy than the singlet at this point (0.68eV) and so a contribution of the triplet state to the magnetic properties would appear negligible in region A (
Figure 7). (3) In region B (
d > 9.5 %) the triplet state is now more stable than the singlet. Thus in this region, a ferromagnetic
Rh-C
60 electronic state is expected arising from flat band ferromagnetism. It is, however, considered that this ferromagnetic phase is different from the experimental ferromagnetic phase since threes calculations reveal a finite band gap (semi-conductor) behaviour whereas the experimental studies indicate metallic behavior. (4) At the point of
d≈9.5%, crossing of the singlet and triplet states occurs.
Figure 7.
S-T gap of Rh-C60 with variation of C60 cage distortion. The information of region and point are in the text.
Figure 7.
S-T gap of Rh-C60 with variation of C60 cage distortion. The information of region and point are in the text.
Around the boundary between regions A and B, the triplet and singlet states are competitive. From the band structure of the singlet state at this boundary, a metallic state is predicted as the ground state, consistent with the experimental study. One possibility to explain the mechanism of ferromagnetism in
Rh-C
60 could be proposed. (1) The experimental ferromagnetic
Rh-C
60 phase occurs around point C, which is found at a distortion
d just below the intersection of the singlet and triplet energies (
Figure 7) which leads to metallic behaviour observed in the experimental ferromagnetic
Rh-C
60. (2) The singlet-triplet gap at the point C is so small that some thermal population of the triplet state may occur. This thermally induced triplet
Rh-C
60 can be considered as an experimentally observed magnetic domain. The population in the excited state at the various temperatures were estimated by the Boltzmann distribution
as illustrated in
Figure 8. The percentage of triplet is expected to be 2% at 300 K with
d = 9.1%. Using this percentage of triplet, the spin concentration value (
n) be estimated as follows:
where V and S denote the volume of primitive cell (=601.8 × 10
-24cm
3) and spin magnitude (=2), respectively. The estimated value is 6.6 × 10
19cm
-3, is comparable with the experimental value (5 × 10
18cm
-3) despite the simplicity of this approximation.
Figure 8.
The proportion of triplet and singlet under the temperature 0~1000K. Each line is obtained from distorted structure d = 4.15, 8.30, 9.14%, respectively.
Figure 8.
The proportion of triplet and singlet under the temperature 0~1000K. Each line is obtained from distorted structure d = 4.15, 8.30, 9.14%, respectively.