Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ
The orbital Knight shifts and g factors for the tetragonal ⁶³Cu₂ + site in HgBa₂Ca₂Cu₃O₈+δ at 133 and 115 K are theoretically investigated based on the high-order perturbation formulae of these quantities for a 3d⁹ ion situated into tetragonally elongated octahedra. The theoretical results reveal go...
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Min-Quan Kuang Shao-Yi Wu Xian-Fen Hu 2017-06-07T15:09:37Z 2017-06-07T15:09:37Z 2014 Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ / Min-Quan Kuang, Shao-Yi Wu, Xian-Fen Hu // Физика низких температур. — 2014. — Т. 40, № 8. — С. 869-872. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS 74.25.N–, 75.10.Dg, 76.30.Fc https://nasplib.isofts.kiev.ua/handle/123456789/119586 The orbital Knight shifts and g factors for the tetragonal ⁶³Cu₂ + site in HgBa₂Ca₂Cu₃O₈+δ at 133 and 115 K are theoretically investigated based on the high-order perturbation formulae of these quantities for a 3d⁹ ion situated into tetragonally elongated octahedra. The theoretical results reveal good agreement with the observed values. The significant anisotropies of the Knight shifts are illustrated as the considerable local tetragonal elongation distortions of the five-coordinated Cu²⁺ sites. The results at different temperatures are also discussed in view This work was financially supported by the Sichuan Province Academic and Technical Leaders Support Fund and the Fundamental Research Funds for the Central Universities under granted No. ZYGX2012YB018. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Свеpхпpоводимость, в том числе высокотемпеpатуpная Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ Article published earlier |
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Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ |
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Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ Min-Quan Kuang Shao-Yi Wu Xian-Fen Hu Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ |
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Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ |
| title_fullStr |
Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ |
| title_full_unstemmed |
Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ |
| title_sort |
theoretical studies of ⁶³cu₂ + orbital knight shifts of hgba₂ca₂cu₃o₈+δ |
| author |
Min-Quan Kuang Shao-Yi Wu Xian-Fen Hu |
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Min-Quan Kuang Shao-Yi Wu Xian-Fen Hu |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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2014 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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The orbital Knight shifts and g factors for the tetragonal ⁶³Cu₂ + site in HgBa₂Ca₂Cu₃O₈+δ at 133 and 115 K are theoretically investigated based on the high-order perturbation formulae of these quantities for a 3d⁹ ion situated into tetragonally elongated octahedra. The theoretical results reveal good agreement with the observed values. The significant anisotropies of the Knight shifts are illustrated as the considerable local tetragonal elongation distortions of the five-coordinated Cu²⁺ sites. The results at different temperatures are also discussed in view
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119586 |
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Theoretical studies of ⁶³Cu₂ + orbital Knight shifts of HgBa₂Ca₂Cu₃O₈+δ / Min-Quan Kuang, Shao-Yi Wu, Xian-Fen Hu // Физика низких температур. — 2014. — Т. 40, № 8. — С. 869-872. — Бібліогр.: 21 назв. — англ. |
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2025-11-26T04:51:50Z |
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© Min-Quan Kuang, Shao-Yi Wu, and Xian-Fen Hu, 2014
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8, pp. 869–872
Theoretical studies of
63
Cu
2+
orbital Knight shifts
of HgBa2Ca2Cu3O8+δ
Min-Quan Kuang, Shao-Yi Wu, and Xian-Fen Hu
School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China
E-mail: mqkuang@yeah.net
Received February 17, 2014, revised March 13, 2014, published online June 23, 2014
The orbital Knight shifts and g factors for the tetragonal
63
Cu
2+
site in HgBa2Ca2Cu3O8+δ at 133 and 115 K
are theoretically investigated based on the high-order perturbation formulae of these quantities for a 3d
9
ion situ-
ated into tetragonally elongated octahedra. The theoretical results reveal good agreement with the observed val-
ues. The significant anisotropies of the Knight shifts are illustrated as the considerable local tetragonal elonga-
tion distortions of the five-coordinated Cu
2+
sites. The results at different temperatures are also discussed in view
of the local structure of the Cu
2+
sites.
PACS: 74.25.N– Response to electromagnetic fields;
75.10.Dg Crystal-field theory and spin Hamiltonians;
76.30.Fc Iron group (3d) ions and impurities (Ti–Cu).
Keywords: g factors, Knight shifts, HgBa2Ca2Cu3O8+δ.
1. Introduction
The Hg-based multilayered cuprate superconductors
have become an important subject due to the unique Jo-
sephson and magnetically couplings [1–3], magnetic [4]
and resistivity and magnetization properties [5,6]. Usually,
the Knight shifts of nuclear magnetic resonance (NMR)
can demonstrate substantial information about the local
structure and electronic properties of the copper-oxygen
planes, which are helpful to the understandings of super-
conductivity and other physical properties of this material.
For instance, the Knight shifts were measured for the te-
tragonal
63
Cu
2+
sites in HgBa2Ca2Cu3O8+δ with the transi-
tion temperatures (Tc) (≈ 133 [7] and 115 [8] K, respec-
tively), and the orbital Knight shifts K and K were
obtained by extrapolation to zero temperature. However,
these experimental results have not been theoretically ex-
plained until now. On the other hand, the simple second-
order perturbation formulae were normally utilized in the
previous studies [9,10] on the Knight shifts of Cu
2+
sites in
some high-Tc superconductors, while the higher (third- and
fourth-) order perturbation contributions were not taken
into account. Moreover, the previous calculations of
Knight shifts did not connect with local structures of the
magnetic sites but induced various adjusted energy separa-
tions.
In order to overcome the above shortcomings in the pre-
vious researches and to study the Knight shifts of
HgBa2Ca2Cu3O8+δ to a better extent, the high-order pertur-
bation formulae of Knight shifts for a tetragonally elongated
3d
9
center are applied in this work. The theoretical calcula-
tions are carried out by correlating with the local structure of
the tetragonal Cu
2+
site using the superposition model.
2. Theory and calculations
The Knight shifts measured at low temperatures [7,8]
can be ascribed to the tetragonal
63
Cu
2+
site in
HgBa2Ca2Cu3O8+δ. This site is coordinated to five oxygen
ions forming a tetragonally elongated octahedron (i.e., one
of the apical ligands of the octahedron is moved to infinity)
[11]. For a Cu
2+
(3d
9
) ion in a tetragonally elongated octa-
hedron, the original cubic
2
Eg ground state may split into
two orbital singles
2
B1g (or |ε >) and
2
A1g (or |θ >). The
former is the lowest level, corresponding to the observed
Knight shifts ( K K [7,8]). Meanwhile, the original
cubic
2
T2g exited state can be separated into an orbital sin-
glet
2
B2g (or | ζ >) and a doublet
2
Eg (or | ξ > and |η >)
[12,13]. As for the previous studies on Knight shifts [9,10],
the orbital contributions from the susceptibility were nor-
mally insufficiently involved, and the third- and fourth-
order perturbation contributions were not considered. Fur-
thermore, the previous treatments failed to establish quan-
titative relationships between g factors and Knight shifts.
To remove the shortcoming of the previous simple se-
cond-order perturbation calculations, the high (third- and
Min-Quan Kuang, Shao-Yi Wu, and Xian-Fen Hu
870 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
fourth-) order perturbation formulae [14] are adopted here
for a tetragonally elongated 3d
9
center. Despite absence of
experimental g factors, the unified studies of Knight shifts
and g factors are performed in this work, since g factors
are also closely relevant to the local structure of the sys-
tem. In view of the related investigations [7–10] on Knight
shifts and g factors, one can obtain the proportionality rela-
tionships between Knight shifts Ki and g shifts (gi – gs,
with i = || and ). Thus, the perturbation formulae of these
quantities can be expressed as follows:
____________________________________________________
2 2 3
2 3 3
2 2 2 2 2 2 2 3
1 1 2 1 22 1 2 2 1 2 1 2 1 2 2
8 4 1 1 4 1 2 1 1 1
2 ,
2 2
s s s
k k k k
g g g k g
E E E E EE E E E E E E E E E E
22 2 3
3
1 2 2 3
2 1 2 2 1 2 2 1 2 2 12 1 2 1 2 2
22 4 2 1 2 1 1 2 1 1 1
,
2 2 2 2
s
s s
gk k k k
g g g
E E E E E E E E E E EE E E E E E
2
2 –3
3 2 2 2 2
1 1 2 1 22 1 2 2
8 4 1 1 4 1
2
2
A B d s
k k k k
K N r g
E E E E EE E E E
2 2
2 2 2 3
1 2 1 2 1 2 2
2 1 1 1
2
2
sk g
E E E E E E E
,
2
2 –3
3
2 1 2 2 1 2 2 1 2 2 1
2 4 2 1 2 1 1 2
2
2
A B d
k k k k
K N r
E E E E E E E E E E E
2
2 2 2 3
1 1 2 1 2 2
2 1 1 1
2 2 2
s
s
g
g
E E E E E E
. (1)
_______________________________________________
In the above expressions, gs (≈ 2.0023) is the pure spin
value, k is the orbital reduction factor, is the spin-orbit
coupling coefficient for the 3d
9
ion in crystals, which is
usually expressed in terms of the corresponding free-ion
value 0 as ≈ k 0; NA is the Avogadro’s number, and μB
is the Bohr magneton. <r
–3
>3d is the expectation value of
inverse cube of the Cu
2+
3d radial wave function.
The denominators E1 and E2 in the above expressions
are the energy separations between the excited
2
B2g and
2
Eg and the ground
2
B1g states, respectively. They can be
written in terms of the cubic field parameter Dq and the
tetragonal field parameters Ds and Dt using the energy ma-
trices for a tetragonal 3d
9
center [15]:
1 210 , 10 – 3 5q q s tE D E D D D , (2)
The structure of the Cu
2+
site in HgBa2Ca2Cu3O8+δ is
characterized by one bond length R along horizontal c
axis and four bond lengths R along vertical a and b axes
[16]: R ≈ 2.741 and 2.786 Å and R ≈ 1.9239 and
1.9273 Å for the Hg-1223 systems with Tc (≈ 135 and
107 K, close to 133 and 115 K, respectively, for those un-
der study). Then the tetragonal field parameters are de-
duced from the superposition model using the local geome-
try of the system [17]:
2 2
2 2 2 /7,
t t
s
R R
D A
R R
4 4
4 16 2 /21 .
t t
t
R R
D A
R R
(3)
Here t2 ≈ 3 and t4 ≈ 5 are the power-law exponents [17],
2A and 4A are the intrinsic parameters, with the reference
distance of the average Cu
2+
–O
2–
bond length:
4 /5.( )R R R R For octahedral 3d
n
ions in com-
pounds, the relationships 4 3 4( )/ qA D and 2 410.8A A
[17–20] have been proved suitable for many systems and
are reasonably adopted here.
From the optical spectral data for Cu
2+
in some oxides
[13], the cubic field parameter Dq (≈ 1300 cm
–1
) and the
orbital reduction factor k (≈ 0.63) can be obtained for the
Cu
2+
site in HgBa2Ca2Cu3O8+δ. The free-ion value of the
spin-orbit coupling coefficient is 0 ≈ 829 cm
–1
[15] for
Cu
2+
. Substituting these values into Eq. (1), the g factors
and Knight shifts are calculated and given in Table 1.
Theoretical studies of
63
Cu
2+
orbital Knight shifts of HgBa2Ca2Cu3O8+δ
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8 871
3. Discussions
From Table 1, one can find that the g factors and
Knight shifts based on the high-order perturbation formu-
lae in this work exhibit reasonable agreement with the ex-
perimental data, and these observed values are also suita-
bly explained in a uniform way.
1. In the present calculations, the theoretical relationships
between Knight shifts (and g factors) and the local structure
of the tetragonal
63
Cu
2+
site in HgBa2Ca2Cu3O8+δ are es-
tablished from the superposition model. So the imperfection
of the previous treatments [9,10] based on the simple se-
cond-order perturbation formulae using various adjustable
energy separations is thus overcome. Notably, the positive
anisotropy 0( )–K K of Knight shifts is consistent with
that 0( )–g g of g factors, with the ratio
– / – 6) %( ) (K K g g due to the linear relationships
between g-shifts and Knight shifts. Moreover, the aniso-
tropies of g factors and Knight shifts can be illustrated as the
local tetragonal elongation of the
63
Cu
2+
site, characterized
by one longer parallel Cu–O bond than the four perpendicu-
lar ones. In fact, the local tetragonal elongation of this site
can be described as the relative elongation ratios
( )– / 31R R R and 33% for the studies systems with Tc of
133 and 115 K, respectively. The highest Tc of the optimally
doped Hg-1223 systems among multilayer cuprate super-
conductors is attributable to the intense hybridization of the
2 2Cu(3 ) O 2 Cu(4( ) )
x y
d p s electron clouds arising
from the long copper-apical oxygen bond lengths R and
flat CuO2 planes and remarkable antiferromagnetic spin
fluctuations in the pairing mechanism [7,8]. In addition, the
theoretical g factors in this work remain to be verified with
further experimental measurements.
2. The g factors and orbital Knight shifts investigated
here correspond to the temperatures slightly lower than Tc
(≈ 133 and 115 K). The quantities used in the calculations
(e.g., the energy denominators E1 and E2, the spin-orbit
coupling coefficient ζ, the cubic field parameters Dq and
the orbital reduction factors k (≈ N)) reflect the local ligand
fields around copper sites and the Cu
2+
–O
2–
orbital admix-
tures (covalency) of the [CuO5]
8–
clusters formed by the
localized Cu
2+
(3d
9
) and the nearest neighbour oxygen lig-
ands with specific local structural data (e.g., the distinct bond
lengths) at the relevant temperatures. So, the above parame-
ters actually contain implicitly the respective contributions at
the corresponding temperatures, and thus temperature T does
not appear explicitly in the detailed calculations.
3. The errors of the present calculations may be dis-
cussed as follows. First, the approximations of the theoret-
ical model and formulae would bring forward some errors.
Second, the errors also arise from the approximation of the
relationship 2 410.8 ,A A which may affect the tetragonal
field parameters and the final results. The errors for the
resultant Knight shifts are estimated to be not more than
3% when the ratio 2 4/A A varies by 10%. Third, the calcu-
lations of this work are performed on the basis of the con-
ventional crystal-field model where the ligand orbital and
spin-orbit coupling contributions are ignored. In view that
the studied Cu
2+
–O
2−
combination has much smaller lig-
and spin-orbit coupling coefficient (≈ 151 cm
–1
[21]) as
compared to that (≈ 829 cm
–1
[15]) of Cu
2+
, the above
ligand contributions can be regarded as very small and
negligible.
4. Conclusion
The g factors and orbital Knight shifts are theoretically
calculated for the tetragonal Cu
2+
sites in the Hg-1223 sys-
tems for temperatures slightly lower than Tc (≈ 133 and
115 K). The significant anisotropies of the Knight shifts are
illustrated as the considerable tetragonal elongations of the
five-coordinated Cu
2+
sites. The higher Tc of the optimally
doped Hg-1223 than other multilayer cuprate superconduc-
tors may be attributed to the stronger hybridization of the
2 2Cu(3 ) O 2 Cu(4( ) )
x y
d p s electron clouds arising
from the longer copper–apical oxygen bond lengths and
flatter CuO2 planes and more significant antiferromagnetic
spin fluctuations in the pairing mechanism of cuprates. It
appears that the present
63
Cu Knight shift analysis can be
helpful to the further understandings of the electronic states
and superconductivity of Hg-1223 systems.
Acknowledgment
This work was financially supported by the Sichuan
Province Academic and Technical Leaders Support Fund
and the Fundamental Research Funds for the Central Uni-
versities under granted No. ZYGX2012YB018.
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63
Cu
2+
sites in HgBa2Ca2Cu3O8+δ
T, K Ref. g g K K
133
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Exp. [7] – – 1.22(2) 0.22(2)
115
Calc. 2.20 2.04 1.14 0.22
Exp. [8] – – 1.16 0.19
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872 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 8
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