Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits
Recently new type of high temperature superconductors is found which are characterized by the existence of circular molecular orbits in each unit site of 2D s/p electron system. In view of the characteristic, a new model of superfluidity is studied based on the coherent state where the zero-point os...
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| Cite this: | Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits / M. Sugahara, N.N. Bogolubov Jr. // Вопросы атомной науки и техники. — 2001. — № 6. — С. 339-342. — Бібліогр.: 12 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-800462025-02-10T01:48:03Z Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits Макроскопическое квантовое состояние и высокотемпературная сверхпроводимость в полулокализованной системе с круговыми молекулярными орбитами Sugahara, M. Bogolubov Jr., N.N. Quantum fluids Recently new type of high temperature superconductors is found which are characterized by the existence of circular molecular orbits in each unit site of 2D s/p electron system. In view of the characteristic, a new model of superfluidity is studied based on the coherent state where the zero-point oscillation of toroidal wave function causes a macroscopic quantum state. This model gives an estimation of the superfluidity transition temperature: Tc≈52-117 K for fcc C60, and Tc≈50-150 K for hole-doped MgB2. 2001 Article Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits / M. Sugahara, N.N. Bogolubov Jr. // Вопросы атомной науки и техники. — 2001. — № 6. — С. 339-342. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 71.10.-w, 71.10.Pm, 73.40.Hm, 74.20.Mn https://nasplib.isofts.kiev.ua/handle/123456789/80046 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Quantum fluids Quantum fluids |
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Quantum fluids Quantum fluids Sugahara, M. Bogolubov Jr., N.N. Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits Вопросы атомной науки и техники |
| description |
Recently new type of high temperature superconductors is found which are characterized by the existence of circular molecular orbits in each unit site of 2D s/p electron system. In view of the characteristic, a new model of superfluidity is studied based on the coherent state where the zero-point oscillation of toroidal wave function causes a macroscopic quantum state. This model gives an estimation of the superfluidity transition temperature: Tc≈52-117 K for fcc C60, and Tc≈50-150 K for hole-doped MgB2. |
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Sugahara, M. Bogolubov Jr., N.N. |
| author_facet |
Sugahara, M. Bogolubov Jr., N.N. |
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Sugahara, M. |
| title |
Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits |
| title_short |
Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits |
| title_full |
Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits |
| title_fullStr |
Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits |
| title_full_unstemmed |
Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits |
| title_sort |
macroscopic quantum state and high-temperature superconductivity in semi-localized 2d electron system with circular molecular orbits |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2001 |
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Quantum fluids |
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https://nasplib.isofts.kiev.ua/handle/123456789/80046 |
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Macroscopic quantum state and high-temperature superconductivity in semi-localized 2D electron system with circular molecular orbits / M. Sugahara, N.N. Bogolubov Jr. // Вопросы атомной науки и техники. — 2001. — № 6. — С. 339-342. — Бібліогр.: 12 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-12-02T14:09:57Z |
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| fulltext |
MACROSCOPIC QUANTUM STATE AND HIGH-TEMPERATURE
SUPERCONDUCTIVITY IN SEMI-LOCALIZED 2D ELECTRON
SYSTEM WITH CIRCULAR MOLECULAR ORBITS
Masanori Sugahara 1 and Nikolai N. Bogolubov (Jr.)2
1 Graduate School of Engineering, Yokohama National University
Hodogaya, Yokohama, 240-8501, Japan
2 V.A. Steklov Mathematical Institute, Academy of Science of Russia
8 Gubkin Str., GSP-1, 117966, Moscow, Russia
Recently new type of high temperature superconductors is found which are characterized by the existence of
circular molecular orbits in each unit site of 2D s/p electron system. In view of the characteristic, a new model of
superfluidity is studied based on the coherent state where the zero-point oscillation of toroidal wave function
causes a macroscopic quantum state. This model gives an estimation of the superfluidity transition temperature: Tc
≈52−117 K for fcc C60, and Tc≈50−150 K for hole-doped MgB2.
PACS: 71.10.-w, 71.10.Pm, 73.40.Hm, 74.20.Mn
I. INTRODUCTION
The cuprate-oxide high Tc superconductivity are
known to be caused by 2D d-electron system in CuO2
network where the probability amplitude of d electron
extends crosswise from each Cu ion. Recently new
trend of high temperature superconductivity [1,2] is
attracting attention where (i) non-d electron system
seems to be responsible to superconductivity, and where
(ii) crystal structure appears to possess ||c 2D network
composed of the inter-connection of circular molecular
orbits. In the report of Schön et al. [1] the surface of
fullerene (C60) fcc crystal was hole-doped by field
effect, where Tc≈52−117 K was found when lattice
constant is 1.415-1.445 nm. The 2D conduction surface
in the system is formed by the inter-connection of the
circular molecular orbit of each spherical C60 molecule.
On the other hand, Nagamatsu et al. [2] found that
MgB2 shows superconductivity with Tc≈39 K. The
crystal structure of MgB2 is composed of 2D network of
B hexagons, which has similarity to the 2D structure of
graphite (2D network of C hexagons) except the
existence of centripetal attraction of electron by the field
of Mg2+ ion.
In this paper we study the macroscopic quantum
state of the 2D electron system in a network which is
composed of the inter-connected molecular units with
circular molecular orbits [3], where each of the 2D
electrons is supposed to be semi-localized in the annular
potential well of the respective molecular orbit [4].
In section II is considered the macroscopic quantum
state originated from the zero-point oscillation of
toroidal wave function. In Section III is given the
estimation of superfluidity threshold temperature of the
materials based on the result of Section II.
II. GROUND STATE OF ONE PARTICLE
AND MANY PARTICLE SYSTEM
We consider a 2D system of charged particle carriers
(charge Q0 and mass M0) in a network, which is
composed of the inter-connected molecular units each
of which has a circular particle orbit with the following
condition. (i) Each of the 2D particles is supposed to be
semi-localized in the annular potential well of the
respective molecular unit. (ii) In the annular well the
ground state wave function ψ0 of each particle has
toroidal amplitude distribution with null angular
momentum (or ψ0(x,y) is real function like a caldera).
(iii) ψ0 makes radial zero-point oscillation of expansion
and contraction by frequency ω (iv) Thermal excitation
of oscillation is forbidden by the condition
ħω>>kBT.
According to the assumption, we select the ground
state wave function of the particle to be a caldera-like
real function
ψ0(m,ζ)=const× mς exp( 2ς /4) with m=1,2,3 ... (1)
where ζ=(x±iy)/l, and l= ω0/ M . The functional
form of (1) is the same as the ground state wave
function for charged particle in magnetic field without
phase factor exp(imθ). The wave function size ml2
may be equated to the effective radius r0 of the
molecular unit. The wave function (1) is found to be the
solution of Hamiltonian
ωω
myxV
yxM
yxH
2
),(
2
,( 22
0
2
−+
∂
∂+
∂
∂−= , (2)
where
22
0
22
22
2
0 1
2
)(
8
),(
yxM
myxMyxV
+
++= ω
(3)
Hψ0(m)=( ћω/2)ψ0(m). (4)
V(x,y) with annular valley is the coulomb potential well
made by the ions of molecular unit (or "lattice"). (2) and
(3) can be rewritten in the following expression using
"vector potential"
−
∂
∂+
−
∂
∂=
2
0
2
0
02
1),( yx AQ
yi
AQ
xiM
yxH
2/),(' ω++ yxV (5)
(Ax,Ay) = (iM0ωx/2Q0, iM0ωy/2Q0) (6)
2
22
0
ˆ
)(2
1),(' S
yxM
yxV
+
= (7)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 339-342. 83
−
∂
∂+
−
∂
∂= yx AQ
yi
yAQ
xi
xiyxS 00),(ˆ
, (8)
Ŝ ψ0(m,ζ)= mћψ0(m,ζ ) m=1,2,3 ... (9)
The vector potential (6) gives the dynamic expression of
the electric force exerted by the lattice when the charged
particle goes outward. In the displacement u from its
equilibrium position, the particle feels lattice charge
density ρ with dielectric constant ε by the equation
0
2
02
2
0 // QuQu
dt
dM εωρερ =−−= ,
ρ makes the following electric field E and vector
potential A
00
2 // QMdivi
t
divdiv ωερω ==−=
∂
∂−= AAΕ
which leads to (6). The physical meaning of (8) and (9)
is explained by the quantization of oscillation energy
exchanged between the particle and the lattice:
x×dpx/dt+ y×dpy/dt
−
∂
∂+
−
∂
∂= yx AQ
yi
yAQ
xi
xi 00
ω
ωm= . (10)
The potential energy (7) expresses the repulsive
polarization potential energy Q0µ/4πεr2, where µ is the
effective electric dipole moment formed in the lattice in
the inward particle motion. The potential may balance
with the kinetic energy
Q0µ/4πεr2 = p2/2M0 = (rp)2/2M0r2 = ћ2m2/2M0r2,
where quantization (10) is considered.
We must note that H in (5) expresses only the
particle energy. The interaction energy ∆E caused by
the exchange of oscillation energy mħω between the
particle and lattice system is estimated using (6) and
(10) as follows.
∆E = - 〉⋅〈 2
0 rAωQ /m ħω = - m ħω/2 = - (ω/2)S. (11)
Now we consider a 2D system of N semi-localized
particles. The ground state ΨN of the system is described
using (1) and (5) as follows:
ENΨN = HNΨN, (12)
HN = ∑ −
N
j jj yxH
1
),( (13)
ΨN = ∏ −
N
j jm
1 0 ),( ςψ (14)
We find the zero-point energy
EN0 = N ħω/2 (15)
and interaction energy by (11)
∆EN = -N mħω/2. (16)
In (13) (xj , yj) are the relative coordinate from the
coordinate (Xj , Yj) of the center of the molecular unit j.
(Xj , Yj) does not appear in the Hamiltonian because
particle energy is independent of it in our assumption. In
case of electron (or fermion) system, one may consider
a Slater determinant in place of (14). In the following
we show that the N-particle Laughlin state ΨLN can have
higher stability than ΨN state employing HN of (13).
It is known that the Laughlin function [5] describes
well the 2D quantum state of fractional quantum Hall
effect (FQHE) [5,6].
)4/exp()(
2
LN ∑∏ −−×=Ψ
> l l
m
kj kjconst ςςς . (17a)
Since the phase of the wave function does not have
direct physical meaning in FQHE [7], we study Ψ′LN
instead of ΨLN
)4/exp()('
2
LN ∑∏ −−×=Ψ
> l l
m
kj kjconst ςςς (17b)
where Ψ′LN is obtainable by a gauge transformation
from ΨLN.
Using the replacement
М ,/
1
Nxx
N
j j
= ∑ −+ Nyy
N
j j /
1
= ∑ −+ , (18a)
,/
1, Npp
N
j jxx
= ∑ −+ Npp
N
j jyy /
1,
= ∑ −+
,(18b)
Nxxx kjjk /)( −= , Nyyy kjjk /)( −= , (18c)
Nppp kxjxjkx /)(, −= , Nppp kyjyjky /)(, −= , (18d)
we can rewrite (17b) into a "diagonalized" form Ψ
)4/exp(
)4/exp('
2
2
LN
∑∏
∑
−×
−×=Ψ
> l lkj
m
jk
l lconst
ςς
ς
(19)
where is used the equality
2
1
22
2
2
1
)()1(1 ∑∑ −−
+−+=
N
j jj
N
j l yx
Nl
Nς
∑
∑∑
>
−−
−+−
+
+
+
=
kj
kjkj
N
j j
N
j j
Nl
yyxx
Nl
yx
2
22
2
2
1
2
1
))()(
∑ >+ +=
kj jk
22 ςς . (20)
Considering (1)~(9), we know that the function (19) is
the zero-point solution of the Hamiltonian
∑ >+ +=
kj jkLN HHH (21)
H+ = H(x+,y+) (S+=0)
Hjk = H(xjk,yjk) (Sjk= ħm)
with eigen-energy
ELN0 =[1+N(N-1)/2] ħω/2 (22)
and interaction energy by (11)
∑ >+ +−=∆
kj jkLN SSE )(2/(ω
= -mN(N-1) ħω/4 (23)
Using (18) and referring to the equality (20) and other
equalities
∑ >++++ +++
kj jkyjkjkxjkyx pypxpypx )( ,,,,
∑ − += N
j jyjjxj pypx1 )(
[m2ћ2/(x2+y2)] ψ0(m,ζ) = )( 22
yx pp +− ψ0(m,ζ).
we find the operator correspondence between (13) and (21)
HLN=HN (24)
with
∑ ∑∑ > −>+ =+
kj
N
j jkj jk SSS
1
ˆˆˆ (25)
84
Based on the correspondence (24) and (25), and
considering the interaction energy, we compare the
system energy between the states ΨN and ΨLN using
(15), (16), (22) and (23)
∆Et = (ELN0)+∆ELN) – (EN0+∆EN )=
= - [(N2 – 3N)(m-1)-2] ħω/4 (26)
∆Et becomes negative when m≥2 and N≥4.
In the hitherto found new type of high temperature
superconductors [1,2], the filling factor νs of carriers per
1 molecular unit may be νs≥1. On the other hand the
ground Laughlin state of the single electron (or fermion)
system appears at the filling factor νs=1/m=1/3,1/5, ... .
Therefore it may be difficult to find there the possibility
of stability of Laughlin state. The situation is different
in case of electron pair (or boson) system, where the
ground Laughlin state appears at the pair filling factor ν
p=1/2, 1/4, 1/6, ... . Suppose an 2D electron system of νs
=1. In pairing the system has pair filling factor νp =1/2
where m=2, and where ∆Et < 0 when N≥4.Of course we
must consider the "Hubbard energy" increase EH ≈ 10eV
per 1 pair. However the energy increase is overcome by
the energy decrease given by (26) when the pair number
Np of the 2D system is large enough to satisfy
Np≥3+4EH/ ħω=Npc. (27)
The pair oscillation frequency is supposed to be ωp=2ω.
When we consider the general rule that the system with
even number of electrons per 1 molecular unit becomes
insulator, we must say that the Laughlin state can be
realized in a 2D system with odd number of electrons
(or holes) per 1 molecular unit if (27) is satisfied.
III. SUPERFLUIDITY
It is known that superfluidity is one of the properties
of the system in the coherent state ΘΨ with definite
phase Θ expressed by
∑ ∞
=Θ ΨΘ=Ψ
0
)exp(
N NN iNw (28)
where NΨ is the particle-number-definite macroscopic
quantum state with N particles, and
wN= 〈N〉N exp(-〈N〉)/N!
where 〈N〉 is the mean particle number. Laughlin state Ψ
′LN is a particle-number-definite macroscopic quantum
state NΨ . Therefore a coherent state may be
composed by the superposition of many Laughlin states
with different N. The least uncertainty condition
∆N∆Θ = ∆p∆r/ħ = 1/2 is satisfied in the coherent state,
realizing the minimization of kinetic and potential zero-
point energy. Therefore we can expect the stable
appearance of a superfluidity based on the coherent state
in the situation where many Laughlin states of νp=1/2
with different pair number Np appear as quantum
fluctuation.
Such a fluctuating situation may be expected in the
multilayer crystal of 2D circular molecular networks
when the filling factor deviates from νp = 1/2 by a small
quantity ∆ν by carrier doping. In case of mono-layer
system, the deviation ∆ν makes quasiparticle excitations
without N fluctuation. In multilayer system, however,
the existence of inter-layer particle exchange leads to
the N fluctuation, the strength of which is determined by
the inter-layer tunneling probability.
It is well known in the experiment of quantum Hall
effect that the localization causes "plateau" where a
Laughlin ground state is stabilized over a finite width of
filling factor. Therefore to observe ideal superfluidity,
one may completely remove the localization using
"ideal" multilayer crystal and set the filling factor in the
de viation νp = 1/2±∆ν. If one increases localization
keeping the filling factor νp = 1/2±∆ν starting from the
ideal state, the macroscopic quantum state may
smoothly change from superfluidity type ΘΨ to
Laughlin type NΨ , so long as the localization
potential is not too strong to destroy even NΨ .
Equating l = r0, we find
ħω = ħ2/M0
2
0r (29)
where r0 is the effective radius of a molecular unit. In
order to consider the thermal effect, we use the thermal
decoherence length
lth = η ħνF/kBT = η(ħ2/kBTM0)(3π2n)1/3 (30)
where η is a coefficient of 1≥η≥1/π. Fermi velocity is
supposed to be given by 3D free carrier model as νF =
(ħ/M0)(3π2n)1/3 for 3D carrier density n. Quantum
coherence may extend over the area lth², where the pair
number is
∆Nth = (1/2π0
2
0r )lth
2 (31)
Supposing the fluctuating appearance of the Laughlin
states when ∆Nth exceed Npc given by (27), we can
determine the threshold temperature of the onset of
superfluidity using (29)-(31)
Tc = (η ħνF/2 π kBr0)/ ω/25.1 HE+ (32)
In fcc C60 with 3 holes in a molecular unit, where we
suppose that 2 holes make filled band and one hole is in
carrier state. Then we find Tc≈70-200 K from (32) with
νF≈4×105m/s r0≈0.5nm and ħω≈0.3eV. The value is in
the same order as Tc≈52-117 K reported in Ref. 1. With
respect to the MgB2, the field made by Mg²+ disturbs the
formation of stable potential valley for π electrons in the
branches of the 2D hexagon network. Concerning hole
carriers on the network, however, the repulsive Mg²+
field makes stable potential valley just on the branch. A
hole (a defect of σ electron) may appear in a hexagon
neighboring the hexagon where new π electron creation
is made through the lowering of π electron potential via
attractive Mg²+ field. In order to reduce zero-point
energy, the hole may take an outside larger orbit around
the hexagon of the new π electron, feeling lattice
potential valley and with some attraction from the newly
created electron. Such hole state may be realized at
auto-doping ratio β = 1/13 (1 hole per 13 hexagons).
Concerning the 2D system of the doped holes, we find ν
F≈4.4×105m/s, r0≈0.62nm, ħω/e≈0.2eV and get Tc≈50-
150 K which is in the same order to the observation
39 K.
85
IV. DISCUSSION AND CONCLUSION
We propose a new model of the high temperature
superconductivity in crystal with 2D plane, which is
composed of the planar connection of circular molecular
orbits. Assuming the semi-localization of carriers in
each molecular unit, we find the following. (i) boson (or
carrier-pair) type Laughlin state with filling factor ν
p=1/2 is the most stable state. (ii) By the superposition
of many Laughlin states, a coherent state with
superfluidity appears in "good" crystal when νp=1/2.
(iii) An estimation of the superfluidity threshold
temperature gives Tc≈70-200 K for fcc C60, and Tc≈50-
150 K for MgB2, which are respectively in the same
order to the reported onset temperature of
superconductivity 52-117 K and 39 K.
It must be noted that there exists some discrepancy
between the assumption used in the model and the
crystal property of the referred experiments. In case of
fcc C60, only the circular molecular orbits (||xy) which
are in parallel with the conduction plane is effective,
and other orbits (||yz and ||zx) have no contribution to the
model. If hole carrier is stabilized in ||xy orbit, electron
carrier must be stabilized in ||yz and/or ||zx orbits. It is
interesting that Tc≈52-117 K is observed in 3 hole
doping per 1 molecule, but lower Tc in 3 electron
doping. Concerning MgB2, we now have not enough
information of its carrier state. Anyway it is difficult in
the attractive field of Mg2+ to consider stable annular
potential well for the graphite-like π electrons existing
on the hexagon network branches. In order to apply our
model, we must suppose the existence of localized holes
at the self-doping ratio β=1/13. However the potential
well of a localized hole has connection paths between
the neighboring wells, which results in the
incompleteness of "semi-localization" condition.
We must also note that the possibility of the paired
superfluid state in FQHE has been theoretically
proposed in the case of the filling factor νS with even
denominator. Greiter et al. [8] discussed the p-wave
pairing between the complex fermions in Pfaffian state
via interaction mediated by vector potential at νS=1/2,
where the solution of the Hamiltonian of the system in
BCS approximation leads to a gap equation having large
pairing energy at relative angular momentum Lz=h. Ho
pointed out [9] that the Pfaffian state and the Ψ331 state
respectively correspond to A phase and A1 phase in
superfluid 3He [10]. Ψ331 may be the ground state of 2
layer FQHE system with νS=1/4 in each layer when the
inter-layer distance ds≅1.5l [11]. Morf pointed out [12]
that the ground state of coulomb-interacting νS=5/2
system is a spin-polarized state with large overlapping
with Pfaffian (or pairing) state.
These pairing states are interesting and deserve
consideration with respect to the electronic state of the
new trend high Tc superconductors in multi-layer crystal
with interconnected networks of 2D circular molecular
orbits. We suppose, however, that the effective filling
factor of the superconductors is νs≅1 which may
exclude the possibility of above pairing states.
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Superconductivity at 52 K inhole-doped C // Nature,
2000, v. 48, p. 549-552; High-Temperature Super-
conductivity in Lattice-Expanded C // Science. 2001,
v. 293, p. 2432-2434.
2. J. Nagamatsu, N. Nakagawa, T. Murakami, Y. Zentani,
and J. Akimitsu. Superconductivity at 39 K in
Magnesium Diboride // Nature. 2001, v. 410, p. 63-
64.
3. M. Sugahara and N. Bogolubov, Jr. High-
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MACROSCOPIC QUANTUM STATE AND HIGH-TEMPERATURE SUPERCONDUCTIVITY IN SEMI-LOCALIZED 2D ELECTRON SYSTEM WITH CIRCULAR MOLECULAR ORBITS
Masanori Sugahara 1 and Nikolai N. Bogolubov (Jr.)2
1 Graduate School of Engineering, Yokohama National University
Hodogaya, Yokohama, 240-8501, Japan
2 V.A. Steklov Mathematical Institute, Academy of Science of Russia
8 Gubkin Str., GSP-1, 117966, Moscow, Russia
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