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PHYSICAL REVIEW B 66, 100501共R兲 共2002兲
Dynamical screening and superconducting state in intercalated layered metallochloronitrides
A. Bill*
Paul Scherrer Institute, Condensed Matter Theory, 5232 Villigen PSI, Switzerland
H. Morawitz
IBM Almaden Research Center, 650 Harry Road, San Jose, California 95120
V. Z. Kresin
Lawrence Berkeley Laboratory, University of California at Berkeley, Berkeley, California 94720
共Received 26 April 2002; published 3 September 2002兲
An essential property of layered systems is the dynamical nature of the screened Coulomb interaction.
Low-energy collective modes appear as a consequence of the layering and provide for a superconductingpairing channel in addition to the electron-phonon-induced attractive interaction. We show that taking into
account this feature allows to explain the high critical temperatures (T c ⬃26 K) observed in recently discovered intercalated metallochloronitrides. The exchange of acoustic plasmons between carriers leads to a significant enhancement of the superconducting critical temperature that is in agreement with the experimental
observations.
DOI: 10.1103/PhysRevB.66.100501
PACS number共s兲: 74.70.Dd, 74.20.Mn
Screening of the Coulomb interaction takes very different
forms in layered conductors and three dimensional 共3D兲 isotropic metals. We show that the dynamic screening in layered
systems can lead to a Coulomb-induced enhancement of the
superconducting pairing and might be an essential addition
to the usual electron-phonon contribution. This important
feature results from the existence of low-energy electronic
collective modes characteristic for layered materials.
The aim of the present paper is to explain the nature of the
superconducting state in layered intercalated metallochloronitrides.1 It has been shown that intercalation of metallic ions and organic molecules into the parent compound
共Zr,Hf兲NCl leads to a superconductor with rather high critical temperature (T c ⬃26 K). 1 Based on experimental
studies1–7 and band-structure calculations8,9 it was concluded
that 共i兲 electron-phonon mediated pairing is insufficient to
explain the high T c ’s observed and that 共ii兲 there is no evidence for the presence of strong correlations; the system can
be described within Fermi-liquid theory. In addition, these
compounds do not have magnetic ions which excludes a
magnetic mechanism as well. No explanation has been suggested so far as to what pairing mechanism can allow to
reach the observed critical temperatures. The theory proposed below shows that such high T c ’s can be obtained by
including the additional pairing contribution arising from the
interaction of carriers with acoustic plasmons; this is the
manifestation of the dynamic screening effect of the Coulomb interaction.
The description of layered conductors can be made by
neglecting the small interlayer hopping in a first approximation. On the other hand, it is essential to take into account the
screened interlayer Coulomb interaction which has an important dynamic part. Indeed, it is known that for usual 3D
materials this interaction can be considered in the static limit
since electronic collective modes are very high in energy 共the
optical plasmon energies are of the order 5–30 eV in metals;
see, e.g., Ref. 10兲. Therefore, the Coulomb repulsion enters
0163-1829/2002/66共10兲/100501共3兲/$20.00
the theory of superconductivity as a single constant pseudopotential ␮ * . The situation is very different in layered conductors: incomplete screening of the Coulomb interaction results from the layering.11 The response to a charge
fluctuation is time dependent and the frequency dependence
of the screened Coulomb interaction becomes important.
This leads to the presence of low-energy electronic collective
modes: the acoustic plasmons. It is this particular feature of
layered materials that brings about an additional contribution
to the pairing interaction between electrons.
The order parameter ⌬(k, ␻ n ) of the superconducting
state is described by
⌬ 共 k, ␻ n 兲 Z 共 k, ␻ n 兲
兺 冕 共 2 ␲ 兲 3 ⌫ 共 k,k⬘ ; ␻ n ⫺ ␻ m 兲 F †共 k, ␻ m 兲 ,
m⫽⫺⬁
⬁
⫽T
d 3 k⬘
共1兲
†
†
c ⫺k,↓
where F † ⫽ 具 c k,↑
典 is the Gor’kov pairing function and
Z(k, ␻ n ) is the renormalization function, defined by
T
Z 共 k, ␻ n 兲 ⫺1⫽
␻n
兺 冕 共 2␲ 兲3
m⫽⫺⬁
⬁
d 3 k⬘
⫻⌫ 共 k,k⬘ ; ␻ n ⫺ ␻ m 兲 G 共 k, ␻ m 兲 .
共2兲
†
G⫽ 具 c k,
␴ c k, ␴ 典 is the usual Green function, and ⌫ the total
interaction kernel; ␻ n ⫽(2n⫹1) ␲ T. We use the thermodynamic Green’s-function formalism 共see, e.g., Ref. 12兲. The
T c for layered superconductors is obtained by solving the set
of Eqs. 共1兲 and 共2兲 self-consistently.
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A. BILL, H. MORAWITZ, AND V. Z. KRESIN
The interaction kernel is composed of two parts, ⌫⫽⌫ ph
⫹⌫ c , where
⌫ ph 共 q; 兩 n⫺m 兩 兲 ⫽ 兩 g ␯ 共 q兲 兩 2 D 共 q, 兩 n⫺m 兩 兲 ,
V c 共 q兲
⌫ c 共 q; 兩 n⫺m 兩 兲 ⫽
.
⑀ 共 q, 兩 n⫺m 兩 兲
2␲e2
␭c
R 共 q 储 ,q z 兲 ⫽
Ṽ 共 q ;q 兲 ,
⑀Mq储
N共 EF兲 c 储 z
⫺ ␮ * ␪ 共 ⍀ c ⫺ 兩 ␻ m 兩 兲 ⫺ ␦ n,m
冎
共3兲
共4兲
共5兲
共6兲
L is the interlayer spacing and ␭ c ⫽e 2 /ប v F 2 ⑀ M is the dimensionless Coulomb interaction constant ( v F is the Fermi velocity兲. The dielectric function ⑀ (q, ␻ n ⫺ ␻ m ) has been calculated for a layered system13–15 in the random-phase
approximation 共RPA兲. It has been shown there that the plasmon spectrum contains anisotropic bands ␻ pl ⫽ ␻ pl (q储 ,q z )
that can be labeled by q z and which are the low-frequency
acoustic modes.
Equations 共1兲 and 共2兲 can be cast into the following matrix form near T c 共see our paper, Ref. 15兲:
兺m 兺 K n,m共 兩 q z兩 兲 ⌽ m共 k z⬘ 兲 ⫽ ␩ ⌽ n共 k z 兲 ,
k z⬘
共7兲
where q z ⬅k z ⫺k z⬘ are the wave-vector components normal to
the conducting layers and ⌽ m (k z⬘ )⫽⌬ m (k z⬘ )/ 冑2m⫹1 is the
reduced order parameter. In the case of a layered superconductor, the matrix K takes the form
K n,m 共 兩 q z 兩 兲 ⫽
1
Nz
再
关 ␭D 共 n⫺p 兲
1
冑2n⫹1 冑2m⫹1
␭ 关 D 共 n⫺m 兲 ⫹D 共 n⫹m⫹1 兲兴
⫹␭ c 关 ⌫ Ic 共 兩 n⫺m 兩 ; 兩 q z 兩 兲 ⫹⌫ Ic 共 兩 n⫹m⫹1 兩 ; 兩 q z 兩 兲兴
共8兲
n⫺m is shorthand for the difference of Matsubara frequencies ␻ n ⫺ ␻ m ⫽2 ␲ T̃(n⫺m) 关with T̃⫽k B T/⍀; we consider
an Einstein phonon ⍀ ␯ (q)⬅⍀]. ⍀ c is the cutoff used to
define the pseudopotential ␮ * , and N z is the number of q z
points considered in the Brillouin zone. All but the static
Coulomb repulsion ␮ * are temperature-dependent quantities.
The critical temperature of the superconducting phase transition T c is reached when the highest eigenvalue is ␩ ⫽1.
⌫ Ic (n, 兩 q z 兩 ) is the frequency-dependent contribution of the
screened Coulomb interaction arising from acoustic plasmons. It has been shown by Morawitz et al.14 that the DOS
of the low-energy collective modes is peaked at q z ⫽ ␲ and
q z ⫽0. Furthermore, it was demonstrated that the q z ⫽0 term
is repulsive and can therefore be included into the pseudopotential ␮ * . 15 The main plasmon contribution to the pairing
is thus obtained for q z ⫽ ␲ /L and has the form
where q 储 (q z ) is the in-plane 共out-of-plane兲 component of the
wave vector. In the last expression, N(E F ) is the 2D electronic density of states 共DOS兲 at the Fermi energy E F , and
Ṽ c (q储 ;q z )⫽R(q储 ,q z )/(2k F q储 ) with
sinh共 q 储 L 兲
.
R 共 q储 ,q z 兲 ⫽
cosh共 q 储 L 兲 ⫺cos共 q z L 兲
兺
p⫽0
⫹␭ c ⌫ Ic 共 兩 n⫺ p 兩 ; 兩 q z 兩 兲兴 .
The first term, ⌫ ph , is the usual pairing contribution resulting from the electron-phonon interaction. D(q,n⫺m)
⫽⍀ 2␯ (q) 关 ␻ n ⫺ ␻ m ) 2 ⫹⍀ ␯2 (q)] ⫺1 is the phonon temperature
Green’s function, and ⍀ ␯ (q) the phonon frequency; summation over phonon branches ␯ is assumed. The second contribution to the interaction kernel, ⌫ c , is the Coulomb part
written in its most general form as the ratio of the bare Coulomb interaction V c (q) and the dielectric function ⑀ (q, ␻ n
⫺ ␻ m ). Both functions have to be calculated for a layered
structure.
The Coulomb interaction for conducting layers separated
by spacers of dielectric constant ⑀ M can be written in the
form11,13
V c 共 q兲 ⫽
2n
⌫ Ic 共 n⫺m 兲 ⫽
冕
␲
␭c
2
q̃ c
0
dq̃
Ṽ c 共 q̃ 兲
冑1⫺q̃ 2 ⑀ 共 q̃,n⫺m 兲
,
共9兲
where q̃⬅q 储 /2k F , and q̃ c ⫽min兵1,兩 ␻ n ⫺ ␻ m 兩 /4E F 其 divides
the ( ␻ ,q) space into the regions ␻ ⬎q v F and ␻ ⬍q v F . The
first region corresponds to the dynamic response and contains plasmon excitations, including the acoustic plasmon
branches. In the second region the response can be treated in
the static approximation and represents the usual repulsive
part of the screened Coulomb interaction. We calculate the
value of the critical temperature from Eqs. 共7兲–共9兲.
In order to demonstrate the importance of dynamic
screening for superconductivity we calculate T c for the following set of realistic parameters: L⫽15 Å, ␭⫽0.5, ⍀
⫽70 meV, ⑀ M ⫽3, v F ⫽5⫻107 cm/s, ␮ * ⫽0.1, and m *
⫽m e . As will be seen below, these values are close to those
found in metallochloronitrides. With these parameters, the
Coulomb interaction constant defined earlier is ␭ c ⫽0.6. One
can, therefore, use RPA in first approximation and neglect
vertex corrections.
With use of aforementioned values for the three quantities
␭, ⍀, and ␮ * one can, in a first step, calculate the value
T c,ph which is the critical temperature in the absence of dynamic screening (⌫ Ic ⫽0). One obtains T c;ph ⫽12 K. If we
now take into account the effect of dynamic screening and
calculate T c using all parameters given above, we obtain
T c ⫽25 K. This demonstrates that the value of T c in layered
superconductors can be drastically affected 共enhanced兲 by
the dynamic part of the screened Coulomb interaction.
We now apply our approach to a specific case among
intercalated metallochloronitrides, namely, the compound
Li0.48(THF) y HfNCl which has a T c ⫽25.5 K. 1 We selected
this compound as a study case because there has been rela-
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PHYSICAL REVIEW B 66, 100501共R兲 共2002兲
DYNAMICAL SCREENING AND SUPERCONDUCTING . . .
tively detailed experimental and theoretical work done on
this layered material. From Refs. 5 and 6 the interlayer distance L and characteristic phonon frequency ⍀ are equal to
L⫽18.7 Å and ⍀⫽60 meV, respectively. The effective
mass and Fermi energy have been estimated from bandstructure calculations.8 Accordingly, m * /m e ⯝0.6 where m e
is the free-electron mass and E F ⯝1 eV. Finally, according
to Ref. 6 we take ␮ * ⫽0.1. Selecting the values ⑀ M ⫽1.8 and
␭⫽0.3 and calculating T c with Eqs. 共3兲–共9兲, we obtain T c
⫽24.5 K, which is close to the experimental value.1 In absence of the plasmon part (⌫ Ic ⫽0) we obtain T c,ph ⫽0.5 K
which indeed confirms that the conventional electron-phonon
mechanism cannot explain the high critical temperature observed in this material.
We point out that the calculation just performed for
Li0.48(THF) y HfNCl makes use of reasonable, but still adjustable parameters ␭ and ⑀ M . A more detailed analysis requires
the experimental determination of these quantities prior to
our calculation. It would thus be of interest to perform tunneling measurements which would allow to determine the
function ␣ 2 (⍀)F(⍀) 关 F(⍀) is the phonon density of states
whereas ␣ 2 (⍀) describes the coupling兴, and correspondingly
␭ 共along with ␮ * ; see, e.g., Refs. 16 and 17兲. Another
method to determine ␭ requires to measure the electronic
heat capacity. Indeed, as is known, the Sommerfeld constant
contains the renormalization factor 1⫹␭ while the magnetic
susceptibility is unrenormalized 共see, e.g., Ref. 17兲. Comparing these two quantities one can extract the value of the
␭
⑀M
T c,ph 共K兲
T c 共K兲
0.5
0.4
0.3
2.2
1.95
1.8
11
4.3
0.5
24.9
25.3
24.6
coupling constant ␭. Such measurements, along with optical
data, would allow to carry out more detailed calculations of
T c for specific metallochloronitrides.
In absence of such experimental data, we present in Table
I a few typical examples of calculated T c for various realistic
values of the parameters ␭ and ⑀ M in Li0.48(THF) y HfNCl.
Note that in all cases the optical plasmon energy at q⫽0 is
of the order ␻ pl,opt (q⫽0)⯝1⫺1.3 eV, in agreement with
band-structure calculation estimates. A more detailed analysis of other metallochlorinitrides will be described elsewhere.
In conclusion, the dynamical screening of the Coulomb
interaction is an essential feature of layered structures that
provides for an additional contribution to the pairing and
leads to a drastic enhancement of T c . The theory presented
here enables us to give an explanation for the high critical
temperatures observed in intercalated layered metallochloronitrides.
10
*Electronic address: abill@psi.ch
S. Yamanaka, K-I. Hotehama, and H. Kawaji, Nature 共London兲
392, 580 共1998兲; S. Yamanaka, H. Kawaji, K-I. Hotehama, and
M. Ohashi, Adv. Mater. 8, 771 共1996兲.
2
H. Kawaji, K. Hotehama, and S. Yamanaka, Chem. Mater. 9,
2127 共1997兲.
3
S. Shamoto, T. Kato, Y. Ono, Y. Miyazaki, K. Ohoyama, M.
Ohashi, Y. Yamaguchi, and T. Kajitani, Physica C 306, 7 共1998兲.
4
S. Shamoto, K. Iizawa, M. Yamada, K. Ohoyama, Y. Yamaguchi,
and T. Katjitani, J. Phys. Chem. Solids 60, 1431 共1999兲.
5
P. Adelmann, B. Renker, H. Schober, M. Braden, and F.
Fernandez-Diaz, J. Low Temp. Phys. 117, 449 共1999兲.
6
H. Tou, Y. Maniwa, T. Koiwasaki, and S. Yamanaka, Phys. Rev.
Lett. 86, 5775 共2001兲; Phys. Rev. B 63, 020508 共2000兲.
7
T. Yokoya, Y. Ishiwata, S. Shin, S. Shamoto, K. Iizawa, T. Kajitani, I. Hase, and T. Takahashi, Phys. Rev. B 64, 153107 共2001兲.
8
R. Weht, A. Filippetti, and W.E. Pickett, Europhys. Lett. 48, 320
共1999兲.
9
I. Hase and Y. Nishihara, Phys. Rev. B 60, 1573 共1999兲.
1
TABLE I. Determination of T c (T c,ph ) in presence 共absence兲 of
the contribution due to dynamic screening of the Coulomb interaction. The parameters are those taken for Li0.48(THF) y HfNCl; ␮ *
⫽0.1, ⍀⫽60 meV, m * /m⫽0.6, E F ⫽1 eV, and L⫽18.7 Å.
P. Nozières and D. Pines, The Theory of Quantum Liquids
共Addison-Wesley, Reading, MA, 1989兲.
11
P.B. Visscher and L.M. Falicov, Phys. Rev. B 3, 2541 共1971兲;
A.L. Fetter, Ann. Phys. 共N.Y.兲 88, 1 共1974兲.
12
A. Abrikosov, L. Gor’kov, and I. Dzyaloshinskii, Methods of
Quantum Field Theory in Statistical Physics 共Dover, New York,
1963兲.
13
V.Z. Kresin and H. Morawitz, Phys. Lett. A 145, 368 共1990兲.
14
H. Morawitz, I. Bozovic, V.Z. Kresin, G. Rietveld, and D. van der
Marel, Z. Phys. B: Condens. Matter 90, 277 共1993兲.
15
A. Bill, H. Morawitz, and V.Z. Kresin, J. Low Temp. Phys. 117,
283 共1999兲; J. Supercond. 13, 907 共2000兲.
16
W. McMillan and J. Rowell, in Superconductivity, edited by R.
Parks 共Marcel Dekker, New York, 1969兲, p. 561; E. Wolf, Principles of Electron Tunneling Spectroscopy 共Oxford University,
New York, 1985兲.
17
G. Grimvall, The Electron-phonon Interaction in Metals 共NorthHolland, Amsterdam, 1981兲, Vol. XVI.
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