Electronic structure of spheroidal fullerenes in a weak uniform magnetic field: a continuum field-theory model

0.11 MB·English

by  M. Pudlak

Electronic structure of spheroidal fullerenes in a weak uniform magnetic field: a
continuum field-theory model
M. Pudlaka, R. Pincaka,b and V.A. Osipovb
aInstitute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47,043 53 Kosice, Slovak Republic
bJoint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Moscow region, Russia
e-mail: [email protected], [email protected], [email protected]
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(February 6, 2008)
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The effect of a weak uniform magnetic field on the electronic structure of slightly deformed
2
fullerene molecules is studied within the continuum field-theory model. It is shown how the exist-

n ing due to spheroidal deformation fine structure of the electronic energy spectrum modifies in the
a presence of themagnetic field. Exact analytical solutions for zero-energy modes are found.
J

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]
i
c
s
I. INTRODUCTION
-
l
r
t
m Recently, we have considered the problem of the low energy electronic states in spheroidalfullerenes [1]. The main
findings were a discovery of fine structure with a specific shift of the electronic levels upwards due to spheroidal
.
t deformation. In addition, three twofold degenerate modes near the Fermi level with one of them being the true
a
zero mode were found. An interesting question is the modification of this structure under the influence of a uniform
m
magnetic field.
-
d The problem of Zeeman splitting and Landau quantization of electrons on a sphere was studied in Ref. [2]. To
n this end, the Schr¨odinger equation for a free electron on the surface of a sphere in a uniform magnetic field was
o formulated and solved. In this paper, we explore the field-theory model suggested in Ref. [1], which describes the
c electronic states near the Fermi energy and takes into account the specific structure of carbon lattice, geometry, and
[
the topological defects (pentagons). The Euler’s theorem for graphene requires the presence of twelve pentagons to

2 get the closed molecule. In the framework of continuum description we extend the Dirac operator by introducing the
v Diracmonopole fieldinside the spheroidto simulate the elasticvorticesdue to twelvepentagonaldefects. The Kspin
4 flux which describes the exchange of two different Dirac spinors in the presence of a conical singularity is included in
1
a form of t’Hooft-Polyakovmonopole. Our studies cover slightly elliptically deformed molecules in the weak uniform
1
external magnetic field.
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0
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0 II. THE MODEL
/
t
a Let us start with writing down the Dirac operator for free massless fermions on the Riemannian spheroid S2. The
m
Dirac equation on a surface Σ in the presence of the abelian magnetic monopole field W and the external magnetic
µ
- field A is written as [3]
d µ
n
iγαe µ[ iW iA ]ψ =Eψ, (1)
o α ∇µ− µ− µ
c
v: where eµα is the zweibein, gµν = eαµeβνδαβ is the metric, the orthonormal frame indices α,β = {1,2}, the coordinate
indices µ,ν = 1,2 ,and =∂ +Ω with
µ µ µ
i { } ∇
X
1
r Ω = ωα β[γ ,γ ], (2)
a µ 8 µ α β
being the spin connection term in the spinor representation (see [1] for details). The energy in (1) is measured from
the Fermi level.
A. Zero-energy mode
Let us start from the analysis of an electron state at the Fermi level (so-called zero-energy mode). We will use the
projection coordinates in the form
1
2R2r 2R2r R2 r2
x= cosφ; y = sinφ; z = c − , (3)
R2+r2 R2+r2 − R2+r2
whereRandcarethespheroidalaxles. TheRiemannianconnectionwithrespecttotheorthonormalframeiswritten
as [4,5]
R2 r2
ω1 =ω2 = − ; ω1 =ω2 =0. (4)
− φ2 φ1 (R2 r2)2+4c2r2 r2 r1
−
The only nonzero component of the gaugepfield W in region R for spheroidal fullerenes reads (see Ref. [1])
µ N
z
W =G+m , (5)
φ
X
where
c2(r2 R2)2+4R4r2
X = − . (6)
r2+R2
p
Notice that the monopole field W in Eq. (5) consists of two parts. The first one comes from the K-spin connection
µ
termandimplies the chargeg = 3/2whilethe secondoneisdueto elasticflowthroughasurface. Thiscontribution
±
is topological in its origin and characterized by charge G. For total elastic flux from twelve pentagonal defects one
has G=1. The parameter m=g G is introduced in Eq.(5). The uniform externalmagnetic field B is chosento be
−
pointed in the z direction so that A~ = B(y, x,0)/2. In projection coordinates the only nonzero component of A
µ
−
is written as
2BR4r2
A = . (7)
φ −(R2+r2)2
Let us study Eq.(1) for the electronic states at the Fermi energy (E = 0). The Dirac matrices can be chosen to be
the Pauli matrices, γ = σ ,γ = σ . By using the substitution
1 2 2 1
− −
ψ ei(j+G)φ u (r)
A = j ,j =0, 1, 2,... (8)
ψB √2π vj(r) ± ±
(cid:18) (cid:19) j (cid:18) (cid:19)
X
we obtain
1 z R2+r2 r2 R2
(j m A )+ (∂ − ) v(r)=0, (9)
r − X − φ K r − 2r(r2+R2)
(cid:18) (cid:19)
1 z R2+r2 r2 R2
(j m A ) (∂ − ) u(r)=0, (10)
r − X − φ − K r − 2r(r2+R2)
(cid:18) (cid:19)
where K = (R2 r2)2+4c2r2. We assume that the eccentricity of the spheroid is small enough. In this case,
−
one can write down c = R+δR and δ ( δ 1) is a small dimensionless parameter characterizing the spheroidal
p | |≪
deformation. Therefore,onecanfollowtheperturbationschemeusingδastheperturbationparameter. Totheleading
in δ approximationEqs. (11) and (12) are written as
(R2+r2)2 r2 R2 (r2 R2)3
∂ v (r)=( [j m(1+δ) − +mδ − ]
r j −r[(R2+r2)2 4R2r2δ] − r2+R2 (r2+R2)3
−
2BR4r r2 R2
+ − )v (r), (11)
−[(R2+r2)2 4R2r2δ] 2r(R2+r2) j
−
(R2+r2)2 r2 R2 (r2 R2)3
∂ u (r)=( [j m(1+δ) − +mδ − ]
r j r[(R2+r2)2 4R2r2δ] − r2+R2 (r2+R2)3
−
2BR4r r2 R2
+ + − )u (r). (12)
[(R2+r2)2 4R2r2δ] 2r(R2+r2) j
−
2
The exact solution to Eqs. (11) and (12) is found to be
v (x)= (x+1)(1/2−m) x+1−2δ−2 δ(δ−1) −α (x+1)2 4δx m, (13)
j x12(j+m+1/2) x+1 2δ+2pδ(δ 1)! −
− −
(cid:0) (cid:1)
p
α
uj(x)=(x+1)(m+1/2)x12(j+m−1/2) x+1−2δ−2 δ(δ−1) (x+1)2 4δx −m, (14)
x+1 2δ+2pδ(δ 1)! −
− −
(cid:0) (cid:1)
p
where x = r2/R2, α = (j/2) δ/(δ 1)+BR2/4 δ(δ 1). The function u can be normalized if the condition
j
− −
m 1/2<j <1/2 mis fulfilled. Inturn, the normalizationconditionforv hasthe form 1/2 m<j <m+1/2.
j
− − p p − −
Finally, there are five normalized solutions for u with j = 0, 1, 2 when m = 5/2, and one solution for v with
j j
± ± −
j =0 when m=1/2. In the limit δ 0 we arrive at the solution for spherical fullerenes
→
vj(r)=r(−1/2−j−m)(R2+r2)(1/2+m)eRB2R+4r2,
uj(r)=r(−1/2+j+m)(R2+r2)(1/2−m)e−RB2R+4r2. (15)
Since normalization conditions do not depend on the parameter δ they are the same as for spherical case. Thus the
perturbation does not change the number of zero modes. Notice that Eq. (15) can be obtained from Eqs. (13) and
(14) by using lim[1+(x/a)]a =ex,a .
→∞
B. Electronic states near the Fermi energy
Let us study now the structure of electron levels near the Fermi energy. It is convenient to consider this problem
by using of the Cartesian coordinates x, y, z in the form
x=a sinθcosφ; y =a sinθsinφ; z =c cosθ. (16)
The Riemannian connection reads (cf. (4))
acosθ
ω1 = ω2 = ; ω1 =ω2 =0. (17)
φ2 − φ1 a2cos2θ+c2sin2θ θ2 θ1
Within the framework of the perturbation scpheme the spin connection coefficients are written as
ω1 = ω2 cosθ(1 δsin2θ), (18)
φ2 − φ1 ≈ −
where the terms to first order in δ are taken into account. In spheroidal coordinates, the only nonzero component of
W in region R is found to be (see Ref. [1])
µ N
W gcosθ(1+δsin2θ)+G(1 cosθ) δGsin2θcosθ, (19)
φ
≈ − −
and the external magnetic field reads
1
A = Ba2sin2θ. (20)
φ
−2
By using the substitution (8) we obtain the Dirac equation for functions u and v in the form
j j
iσ 1(∂ + cotθ)+ σ2 j mcosθ+ 1Ba2sin2θ +δˆ uj(θ) =E uj(θ) , (21)
− 1a θ 2 asinθ − 2 D1 vj(θ) vj(θ)
(cid:18) (cid:18) (cid:19) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
where
γ Ba
ˆ = 1 sinθ(j 2mcosθ) γ sin3θ. (22)
1 1
D − a − − 2
3
A convenient way to study the eigenvalue problem is to square Eq. (21). For this purpose, let us write the Dirac
operator in Eq. (21) as ˆ = ˆ +δˆ . One can easily obtain that (similar way as in Ref. [6])
0 1
D D D
1 cosθ 1 1 (j mcosθ)2 B
ˆ2 = ∂2+ ∂ + − +Bj+ (σ 2m)cosθ
D0 −a2 θ sinθ θ− 4 − 4sin2θ a2sin2θ 2 3−
(cid:18) (cid:19)
m jcosθ B2a2sin2θ
+σ − + . (23)
3 a2sin2θ 4
NoticethattofirstorderinδthesquareoftheDiracoperatoriswrittenas ˆ2 =(ˆ2+δΓˆ),whereΓˆ =(ˆ ˆ + ˆ ˆ ).
D D0 D0D1 D1D0
In an explicit form
a2Γˆ =2j2+jx(σ 6m)+4m(m σ )x2+2mσ +3Ba2(1 x2)x(σ /2 m)
3 3 3 3
− − − −
+2Bja2(1 x2)+B2a4(1 x2)2/2 (24)
− −
where the appropriate substitution x=cosθ is used. The equation ˆ2ψ =E2ψ takes the form
D
[∂ (1 x2)∂ (j−mx)2−jσ3x+ 41 +σ3m a2BV(x) δa2Γˆ] uj(x)
x − x− 1 x2 − − vj(x)
− (cid:18) (cid:19)
1 u (x)
= (λ2 ) j , (25)
− − 4 vj(x)
(cid:18) (cid:19)
where λ=aE,V(x)=j+(σ 2m)x/2. Since we consider the case of a weak magnetic field the terms with B2 and
3
−
δB in Eq. (25) can be neglected. As a result, the energy spectrum for spheroidal fullerenes is found to be
(λδ )2 =(n+ j +1/2)2 m2+Ba2j+Ba2A +δF(j,n,m) (26)
jn | | − jn
where
j(m2 1/4)
A = − .
jn
− p(p+1)
The function ofspheroidaldeformationsF(j,n,m)is specifiedin Ref.[1]. Finally,in the linear inδ approximation,
the low energy electronic spectrum of spheroidal fullerenes takes the form
Eδ =E0 +E0Bz +Eδ (27)
jn jn jn jn
with
Ba2(j+A ) δF(j,n,m)
E0 = (2ξ+n)(2η+n), E0Bz = jn ,Eδ = ,
jn ± jn 2E0 jn 2E0
jn jn
p
where ξ =µ (ν) and η =β (α) for j >0 (j <0), respectively. Here we came back to the energy variable E =λ/a (in
units of h¯V /a where V is the Fermi velocity).
F F
As is seen from Eq. (27), for B = 0 the spheroidal deformation gives rise to an appearance of a additional
structure. As an example, Table 1 shows all five contributions (in compliance with Eq. (27)) to the first and second
energy levels for YO-C (YO means a structure given in [7,8]). As is seen, the energy levels become shifted due
240
to spheroidal deformation. The uniform magnetic field provides the well-known Zeeman splitting. The difference
betweentopologicalandZeemansplitting isclearlyseen. Inthe secondcase,the splittedlevelsareshiftedinopposite
directions while for topological splitting the shift is always positive. As an illustration, Fig. 1 schematically shows
the structure ofthe secondlevel. Inthis case,the initial (for δ =0)degeneracyofE0 is equalto six. The spheroidal
jn
deformation provokes an appearance of three shifted double degenerate levels (fine structure). The magnetic field is
responsible to Zeeman splitting.
We admit that the obtained values of the splitting energies can deviate from the estimations within some more
precise microscopic lattice models and density-functional methods (see, e.g. [9]). In our paper, we focused mostly on
theveryexistenceofphysicallyinterestingeffects. Forthisreason,thevaluesinbothTableandthe schematicpicture
are presented with accuracy at about one percent of E0 .
jn
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III. CONCLUSION
In conclusion, we have considered the structure of low energy electronic states of spheroidal fullerenes in the weak
uniform magnetic field provided the spheroidal deformation from the sphere is small enough. For the states at the
Fermi level, we found an exact solution for the wave functions. It is shown that the external magnetic field modifies
the densityofelectronicstatesanddoes notchangethenumber ofzeromodes. Fornon-zeroenergymodes,electronic
statesneartheFermienergyofspheroidalfullerenearefoundtobesplittedinthepresenceofaweakuniformmagnetic
field. It should be mentioned that the zero-energy states in our model corresponds to the HOMO (highest occupied
molecularorbital)statesinthecalculationsbasedonthe local-densityapproximationinthedensity-functionaltheory
(see,e.g.[10]). Inparticular,the HOMO-LUMOenergygapisfoundto be about1.1eVfor YO-C fullerene within
240
our model (here LUMO means the lowest unoccupied molecular orbital).
The work was supported in part by VEGA grant 2/7056/27. of the Slovak Academy of Sciences, by the Science
and Technology Assistance Agency under contract No. APVT-51-027904 and by the Russian Foundation for Basic
Research under Grant No. 05-02-17721.
[1] M. Pudlak, R.Pincak and V.A. Osipov, Phys.Rev.B 74 (2006) 235435.
[2] H.Aoki and H.Suezawa, Phys. Rev.A 46 (1992) R1163.
[3] N.D.Birrell and P.C.W. Davies, Quantum Fields in Curved Space, (Cambridge 1982).
[4] M. Nakahara, Geometry, Topology and Physics, (Instituteof Physics Publishing Bristol 1998).
[5] M.G¨ockeler and T.Schu¨cker, Differential geometry, gauge theories, and gravity (Cambridge UniversityPress 1989).
[6] D.V.Kolesnikov and V.A. Osipov,Eur.Phys.Journ. B 49 (2006) 465.
[7] M. Yoshida and E. Osawa, Fullerene Sci. Tech. 1(1993) 55.
[8] J. P. Lu and W. Yang, Phys.Rev. B 49 (1994) 11421.
[9] R.A.Broglia, G. Col`o, G. Onida, H.E. Roman, Solid State Physics of Finite Systems, (Springer2004).
[10] S.Saito and A.Oshiyama, Phys. Rev.Lett. 66 (1991) 2637.
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YO−C240 Ba2 =0.1 j Ej0n(eV) Ej0nBz(meV) Ejδn(meV)
n=0, m=1/2 1 1.094 27 10.5
-1 1.094 -27 10.5
n=1, m=1/2 1 1.89 16 8.8
-1 1.89 -16 8.8
n=0, m=1/2 2 1.89 32 28.4
-2 1.89 -32 28.4
n=0, m=−5/2 3 1.89 24 3
-3 1.89 -24 3
TABLEI. The structure of the first and higher energy levels for YO-C240 fullerene in uniform magnetic field. The hopping integral
and other parameters are taken to be t = 2.5 eV and VF = 3tb/2¯h, b= 1.45˚A, R= 7.03˚A, SD = 0.17˚A, δ = 0.024. b is average bond
length,R(R=a)isaverageradius,SD isstandarddeviationfromaperfectsphere(seeRefs. [7,8]),sothatδ=SD/R.
E0 +Eδ +E0Bz
jn jn jn
FIG. 1. The schematic picture of the second positive electronic level Eδ of the spheroidal fullerenes in a weak uniform
jn
magnetic field.
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