Calculated superconductive properties of Li and Na under pressure
N.E. Christensen1 and D.L. Novikov
1
2
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark and
2
TIAX LLC, Acorn Park, Cambridge, Massachusetts 02140-2390, USA
(Dated: May 5, 2006)
Using ab initio calculations of electronic structures and electron-phonon coupling in each phonon
state we examine the superconductive properties of Li and Na under pressure. From the Eliashberg
equations it is found that a Coulomb pseudopotential parameter (µ∗ ) of ”usual” magnitude (0.13)
yields Tc values for fcc-Li close to experiments, and they also agree well with recent similar calculations by Tse et al. (J. Phys. Cond. Mat. 17, S911 (2005)), Maheswari et al. (J. Phys. Soc.
Jpn. 74, 3227 (2005)), and Kasinathan et al. (Phys. Rev. Lett. 96, 047004 (2006)). Consequently,
the Tc values for fcc-Li predicted earlier by us (Phys. Rev. Lett. 86, 1816 (2001)) are clearly too
high, and reasons for this are discussed. The calculations for sodium suggest that superconductivity
should not be observed in the fcc phase, except, maybe very close to the fcc-cI16 transition, but
then with Tc at most ∼1 K. Both fcc-Li and fcc-Na become dynamically unstable (soft modes) as
the pressure exceeds ∼ 40 and ∼ 90 GPa, respectively.
PACS numbers: 71.20.Dg,74.10+v,64.30.+t,74.62.Fj
I.
INTRODUCTION
The alkali metals transform from simple structures,
mostly the body-centered cubic (bcc) form to sequences
of new, often complex, structures as pressure is applied.
Some of the high-pressure phases are low-coordinated
and characterized by multicenter bonding. The lightest
alkali elements, lithium and sodium, have been studied
both theoretically, see for example Refs. 1–9 and experimentally, Ref.2. At low temperature both Li and Na undergo a martensitic transition to the 9R (samarium-type)
structure, but at higher temperatures (∼ 180 K) lithium
undergoes2 the transition sequence bcc→fcc→hR1→cI16
as an applied pressure pressure is increased from 0 to 50
GPa. The cI16 structure is a distorted 16-atom bcc superstructure (SG 220), observed for the first time for an
element by Hanfland and Syassen.2 Sodium follows a similar pattern, apart from the fact that the hR1 (rhombohedrally distorted f cc) was not observed. Experimentally10
the onset of Na-cI16 was found at a much higher pressure (103 GPa at room temperature) than in Li (≈40
GPa). For Li our calculated2,4 transition pressures agree
very well with the experimental values, although we did
not include phonon contributions to the free energy. For
sodium we found4,11 a transition from fcc to cI16 at
130 GPa, i.e. substantially higher than observed. The
transition pressure obtained theoretically by Neaton and
Ashcroft3 is also 130 GPa. Although differences between
theory and experiment can be expected, we still believe
that the discrepancy in this case is significant, and it will
be discussed later.
Both the structural change upon cooling at zero pressure as well as the surprising pressure induced structural
transformations observed, and still to be observed, indicate that these materials which might appear ”simple” at ambient conditions, are not at all simple when
their equations of state are examined over large ranges
of thermodynamical parameters. It has for some time
been a puzzle that Li is not found to be a superconductor (at ambient pressure), in particular since application
of the BCS theory would suggest a transition temperature around 1 K.12 Experiments showed that Li is a
normal metal down to at least 6 mK.13 Liu and Cohen12
examined whether the discrepancy between earlier calculations, which were performed for the bcc phase, and
experiment could be caused by the electron-phonon (ep) coupling being weaker in 9R-Li than in bcc-Li. They
calculated the e-p matrix elements and determined the
coupling constant λ. No significant difference between
the λ values for bcc- (0.51) and 9R-Li (0.41) was found,
but the authors note that only very few phonon modes
could be sampled due to lengthy calculations. Using the
McMillan equation:14
Tc =
n −1.04(1 + λ) o
<ω>
,
exp
1.2
λ − µ∗ (1 + 0.62λ)
(1)
with the Coulomb pseudopotential16 parameter µ∗ =
0.12, and the averaged phonon frequencies < ω > equal
to 180 and 200 K, the Tc values 1.73 and 0.58 K were
found12 for bcc and 9R, respectively. For 9R Li this
is clearly too high compared to the experimental upper limit, 6 mK. Later, however, Liu et al.15 performed
new, refined calculations within the Eliashberg17,18 theory and found larger differences between the electronphonon coupling in the two structures. This appears to
explain why Li in the low-temperature phase is not superconducting, or has a vanishingly small Tc value. As
illustrated in Fig. 3 in Ref. 15, the electrons in bcc-Li
have substantially stronger coupling to the low frequency
phonon modes than is found in 9R. This explains why Tc
is much higher in bcc-Li. In Ref. 19 some of the same
authors estimate the maximum value of µ∗ to be 0.23 for
Li from the Anderson-Morel relation,
µ∗max = 1/ln(EF /ΘD ),
(2)
2
80
C-178-542A
Li-fcc
RM TA + M cM illan
*= 0
0
0.2
5
0.2
0
3
.
0
5
3
0.
60
Tc (K)
.13
40
20
0
0
20
40
60
P (GPa)
FIG. 1: Tc calculated for Li-fcc vs. pressure for various choices
of µ∗ . These calculations within the Rigid Muffin Tin Approximation, in Ref. 20, just shown here as functions of pressure.
These Tc values turned out to be much higher than those
measured later, Refs. 22 and 23, which did not exceed 20 K.
where EF and ΘD are the Fermi and Debye temperatures. Calculating Tc vs µ∗ it was concluded that with
µ∗max ∼ 0.2 Tc is larger than 1 mK for 9R Li and larger
than 0.1 K for bcc Li (insert in Fig. 1 of Ref. 19). This
agrees very well with our conclusions in the following
analysis of the e-p coupling and superconductivity in the
high-pressure phases.
Earlier, we presented20 calculations of Tc for Li-fcc under pressure using Eq.1 with phonon parameters obtained
essentially from ab initio supercell calculations. The e-p
coupling parameter, λ, was calculated within the Rigid
Muffin-Tin Approximation, RMTA.21 Later, it was experimentally confirmed22–24 that Li does become a superconductor under pressure, but the calculated20 Tc values
were by far too high, at least unless very large values of
µ∗ were assumed. This is illustrated in Fig. 1, where we
show the old calulations vs. pressure for µ∗ values ranging from 0.13 to 0.35. A ”standard” value is µ∗ ∼ 0.13.
It was concluded20 that superconductivity occurs in
fcc lithium as a consequence of softening of the lattice
with pressure, clearly seen in the volume dependence
of phonon frequencies and elastic shear constants. The
fact that we found Tc values which turned out to be
by far too large, indicates that there are features of the
model, like our implementation of the RMTA, which are
not well suited to calculations for (fcc) lithium. We
therefore investigated whether a direct solution of the
Eliashberg equations would lead to more reasonable results for ”standard values” of µ∗ . This turned out25,26
to be the case, and this was also shown by similar published results by Tse et al.27 , Maheswari et al.28 , and
very recently by Kasinathan et al.29 Although the latter
publications demonstrate that the detailed calculation of
electron-phonon coupling in each state and the use of
the Eliashberg equations lead to results which agree well
with experiments, we still wish to present our results and
discuss what went wrong in our RMTA calculations20 for
Li. This analysis is also of interest since RMTA calculations of Tc , which appear to agree quite well with experiments, have been published later.30,31 Also, during the
past, the RMTA has been applied sucessfully to many
transition metals, and it would require very strong arguments to classify all these calculations as fundamentally
wrong. We shall point out that there are cases, including
lithium, for which the averaged electron-phonon interaction implied in the RMTA is not a good description. A
few phonon states states with low frequencies may exhibit a particularly strong coupling to the electrons, and
their contribution will dominate. These generate low averages of phonon frequencies, and consequently a large λ
(e-p coupling). Although the basic reason for a high Tc
was mentioned in Ref. 20, the effect could not be correctly described within the RMTA. The fact that we still
got too high Tc values was caused by an ambiguity in (at
least) our RMTA implementation, as discussed later.
Further, we also wish to examine whether sodium, in
the fcc structure, becomes a superconductor at high pressures. Using exactly the same method as for Li in Ref. 20
this would appear to be the case25 , but that conclusion
is not supported by the the more elaborate calculations.
All calculations referred to so far, including ours described in the following, have one serious problem in
common, namely the treatment of the electron-electron
repulsion by a single adjustable parameter, the MorelAnderson pseudopotential. The validity of this method
has been questioned for metals like Li by Richardson and
Ashcroft.32,33 However, an important development was
made by Gross and his coworkers,34 who developed an
ab-initio density-functional theory for superconductivity
without the µ∗ parameter. This theory was recently35
applied to lithium and other metals. The Tc values obtained for fcc-Li agree well with experiments.
The remaining part of this paper contains two sections.
The next one describes briefly our method of calculation
and presents the results which differ significantly from
those represented in Fig. 1. The third section contains
our summary and conclusions.
II.
METHOD AND RESULTS
This section briefly describes the methods used to perform the detailed calculations of the electron-phonon interaction, using linear response theory and solving the
Eliashberg equations. These considerations are contained
in subsection A, which also presents the results obtained
for lithium. The results of the calculations for sodium
are given in subsection B.
3
A.
Method and Results for Li
C-178-540
Tc =
n −1.04(1 + λ) o
f1 f2 ωlog
,
exp
1.2
λ − µ∗ (1 + 0.62λ)
ωlog = exp
λ
∞
0
o
dω 2
α F (ω) ln ω .
ω
Z
∞
0
dω 2
2α F (ω).
ω
(4)
(5)
Before discussing the results obtained for Tc vs pressure for lithium, we apply Eq. 2 to estimate an upper
limit for µ∗ . Figure 2 shows how the Fermi level varies
with volume in Li-fcc. The Debye temperature is estimated from
ΘD = 1.4
~ √
< ω 2 >,
kB
(6)
where the prefactor, 1.4, is semi-empirical, kB is Boltzmann’s constant, and < ω 2 > is obtained from:
< ω 2 >=
2
λ
Z
∞
dωωα2 F (ω).
fcc-Li
3.8
3.7
3.6
In this equation α2 F (ω) is the Eliashberg function, α(ω)
the frequency (ω) dependent electron-phonon coupling,
and F (ω) the density of phonon states. The over-all e-p
coupling parameter, λ, is obtained from:
λ=
3.9
(3)
where the prefactors f1 and f2 (close to unity in our
cases) are defined in Ref. 40 (eqs. (35) and (36)), and
ωlog is given by:
n2 Z
4.0
EF (eV)
The phonon states in Li and Na under pressure were
calculated by means of the linear-response method as implemented in a full-potential LMTO (Linear Muffin Tin
Orbital36 ) scheme by Savrasov.37 For each phonon mode
and wavevector the e-p coupling was calculated as described by Savrasov and Savrasov.38 Subsequently, the
Eliashberg equations were solved, and as results we obtained the Tc value corresponding to a chosen value of µ∗ ,
or, alternatively, the µ∗ value, which leads to a selected
value of Tc . Further, we examine in more detail the effects of varying µ∗ by applying the McMillan equation as
modified by Allen and Dynes:39,40
(7)
0
It follows from Fig.3 that µ∗max for Li-fcc is ≈ 0.22 over
the entire volume range considered here. This agrees with
the estimate in Ref. 19 for Li at ambient pressure. The
calculated Tc values for Li-fcc under pressure are shown
in Fig. 4 and compared to experimental data obtained
in Refs. 22 and 23. Also, the early tentative assignment
made by Lin and Dunn41 of an observed conductivity
anomaly to superconductivity, is marked. The very nice
3.5
0.3
0.4
0.5
0.6
0.7
V/V0
0.8
0.9
1.0
FIG. 2: Fermi level for Li-fcc vs. volume. V0 is the volume
at ambient pressure. (Energy zero at the band bottom at Γ.)
experimental data obtained in Ref. 24 were not included
in this figure, but we note that Ref. 24 finds that Tc
reaches a maximimum as the pressure reaches the upper
limit for stability of the Li-fcc,2 and that it decreases in
cI16-Li. This was also predicted in Ref. 20. There are no
experimental data for P< 20 GPa in Fig. 4, and Ref. 24
demonstrates that no superconductivity is found in this
low-pressure regime. The calculations for fcc-Li do not
reproduce this behavior, and probably the phase diagram
at low T differs from the measurements2 at 180 K.
A comparison of the full-line curve with the one shown
dotted in Fig. 4 indicates that, for a given µ∗ the AllenDynes formula, Eq. 3, matches the results obtained
directly from the Eliashberg equations within ≈ 1 K.
The prefactors, f1 and f2 , used in Eq. 3 were 1.0706
and 1.0338, respectively, at V/V0 =0.5 (P= 26 GPa) and
1.0111, 1.0020 at ambient pressure (V/V0 =1.0). Thus,
omitting these factors (setting them equal to 1) would
reduce Tc by ≈ 10 % at 26 GPa.
The present calculations of Tc for fcc-Li, combining
linear response phonon calculations and the solution of
the Eliashberg equations, yield values which are close to
experiments, if we use µ∗ around 0.20. Thus, it is not
necessary to invoke the very large µ∗ values which according to Fig. 1 would be needed to bring the previous calculations20 into the range of the experimental Tc
values. The question is then: ”Why do the results obtained here differ so much from those of Ref. 20 ?” The
McMillan and the Allen-Dynes equations are very similar, Eqs. 1 and 3. The latter even includes the Tc enhancing prefactors, f1 and f2 . The explanation follows by
4
0.25
m ax
(K)
C-178-541
0.20
b)
comparing the key parameters as listed in Tables I and II
just for two pressures, 14 and 26 GPa. Table I (labelled
”RMTA”)
lists η (the Hopfield parameter), λ, < ω >,
√
and < ω 2 > as calculated in Ref. 20, whereas Table
II (”Eliashberg”) contains the corresponding parameters
obtained here, and to be used in Eq. 3.
0.15
TABLE I: RMTA: Li-fcc parameters used in Ref. 20 in the
McMillan equation.
a)
D
(K)
450
400
V/V0
0.5
0.6
fcc-Li
P (GPa) η (eV/Å2 ) λ
26
3.93
1.42
14
2.36
1.00
< ω > (K)
451
416
√
< ω 2 > (K)
473
436
350
300
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
TABLE II: Eliashberg: Li-fcc parameters used in the AllenDynes equation, Eq. 3.
V/V0
FIG. 3: a) Debye temperature (Eq. 6), and, b), the maximum
value of µ∗ as estimated from the relation 2.
20
C-178-545
fcc-Li
E
*=0.13
0.
15
Tc (K)
15
10
E ( *= 0.24)
20
0.
5
0
0
10
20
P (GPa)
30
40
FIG. 4: Tc of Li-fcc vs. pressure. The experimental results
included are: Filled circles: Ref.23, diamonds: Ref.22, and
triangle: Ref.41. Present calculations (open circles): Full
line curve: Directly from Eliashberg equations using µ∗ =0.13.
Dotted, dashed, and dash-dotted curves: From Eqs. 3 to 5
and choosing µ∗ =0.13, 0.15, and 0.20. A single point marked
”E” represents the case where Tc =11 K is inserted in the
Eliashberg equations at the volume V=0.5V0 (i.e. 50 % compression relative to equilibrium, P= 25.9 GPa). The µ∗ calculated with these settings was 0.24. Note that there are no
experimental data below P= 20 GPa. See also Ref. 24.
V/V0
0.5
0.6
P (GPa) η (eV/Å2 ) λ
26
1.19
1.37
14
0.89
0.89
ωlog (K)
191
222
√
< ω 2 > (K)
276
296
It is seen that phonon frequency averages differ significantly. The < ω > values used in the RMTA calculations
are almost twice the ωlog deduced
from Eq. 4 and a sim√
2
ilar remark applies to the < ω > values used in the
two approaches. Although Ref. 20 used ab initio supercell calculations to scale the phonon frequencies it was
not possible to include correctly the increased weighting
of the e-p coupling at low frequencies with pressure which
is characteristic to the alkali metals. In the simpler calculation equations 4 and 7 could not be used since the
Eliashberg functions were not available. It is interesting
to note, that the λ values are more or less the same in the
two tables (and this is also the case for the whole pressure
range). Then it would appear that the only important
difference between the calculations of Ref. 20 and the
present is the values of < ω > used in Eq. 1 and ωlog
used in Eq. 3, and that the RMTA as such is sufficiently
accurate in our case. However, this is not quite correct.
The Hopfield parameters (η= λM < ω 2 >, M being the
ionic mass) are very different. In the ”Eliashberg case”
λ is calculated from the Eliashberg equations, whereas it
in the RMTA is obtained from η. Since the η value is
very large in the RMTA, and < ω 2 > also too large, λ
gets a reasonable value due to cancellation of errors.
It remains to be examined why the η values obtained
in Ref. 20 were too large, apparently. The expressions
for η, see Refs. 21,42–44, contain total as well as partial density-of-states (DOS) functions. The partial DOSs
where obtained by angular-momentum projections onto
muffin-tin spheres only, and therefore quite significant
contributions corresponding to the interstial regime are
5
1.0
1.0
C-178-466/465
fcc-Li
V/V0=1.00
C-178-468/467
fcc-Li
a)
V/V0 =0.50
0.8
0.8
0.6
F ( )
0.6
2
F ( )
a)
2
0.4
0.4
0.2
0.2
0.0
0.0
fcc-Li
fcc-Li
b)
V/V0=1.0
0.8
V/V0 =0.50
3
(THz)
0.6
2
2
(THz)
b)
2
0.4
1
0.2
0.0
0
2
4
6
8
10
f (THz)
FIG. 5: Li-fcc at V=V0 : a): Eliashberg function, and b): α2
vs. phonon frequency, f.
missing. (In fact a projection onto Wigner-Seitz spheres,
which are larger, would have been more appropriate).
The total DOS can be obtained accurately directly from
the band structure in the standard way, but we decided
to calculate this as a sum of the partial DOSs, the reason being that this would to some degree compensate
for the errors caused by the underestimated partial functions. For volumes near 0.4×V0 the total DOS (at the
Fermi level) is 50-60 % larger than what is obtained by
the summation mentioned. It might be suggested then,
that our η values should be multiplied by a factor 1/1.6,
approximately. Using a reduction factor 1/1.58, the old λ
values for fcc-Li for P= 26 and 14 GPa would be reduced
to 0.89 and 0.62 instead of the values cited in Table I,
and with µ∗ = 0.13 the corresponding Tc values would
be 21.6 and 7.9 K, respectively. In fact the ”theoretical” transition temperatures would have been in excellent
agreement with the later experiments.22,23 However, this
agreement is obtained for the wrong reason. The λ values
are now too small because the < ω 2 > used is too large,
but the too large value of the prefactor ”accidentally”
compensates for this error. A similar problem is encountered in the RMTA calculations by Iyakutti and Louis.31
The λ values are too small, and averages of phonon fre-
0
0
5
10
15
20
25
f (THz)
FIG. 6: Li-fcc at V=0.5V0 (P=26 GPa): a): Eliashberg function, and b): α2 vs. phonon frequency, f.
quencies too high, Table I of Ref. 31. Thus, for Li under
pressure we see no way to obtain quantitative agreement
with experiments using the RMTA with its implicit averaging of the electron-phonon coupling. In the cases
where the numbers do agree, this is due to a fortuitous
cancellation of errors.
The reason for occurrence of superconductivity in Li
is an increase in coupling between electrons and lowfrequency phonon modes as pressure is applied. Simultaneously, this also explains why ωlog and < ω 2 >
as obtained by Eliashberg averaging, i.e. using the
weight function g(ω) = (2/(λω))α2 F (ω), are significantly
smaller than those calculated with a usual normalized
phonon density of states as weight function (Table I).
The two figures, 5 and 6, show the Eliashberg function,
α2 F (ω), and the average e-p coupling, α2 , as functions
of f = ω/2π at the zero-pressure volume, V0 , and for
for the 50 % compressed fcc lithium crystal. The figures clearly illustrate the increase in electron coupling to
low-frequency modes with pressure.
6
C-178-456
15
2.0
N a -fc c s u p e rc e ll, G G A
Na-fc
1.5
f ( T H z)
10
1.0
T
C
(K)
LA
5
0.5
TA
0
0 .2
0.0
0 .4
0 .6
V /V 0
0 .8
FIG. 7: Frequencies of the longitudinal (LA) and transversal
(TA) acoustic modes at the symmetry point X in the Brillouin
zone of Na-fcc vs. volume. As the transition to the cI16
structure is approached (V∼0.28V0 , P∼106 GPa, theory) a
softening of the lattice occurs in the transverse modes. The
lowest (T1 ) transversal modes for wavevectors along (110) go
soft at a lower pressure, ≈ 90 GPa.
B.
0
1.0
Sodium.
The maximum value of µ∗ was estimated (Eq. 2) as for
lithium, and also in Na it is ≈ 0.22. As for lithium, the
fcc-structure softens as the external pressure is increased
on sodium. This is illustrated in Fig. 7 where the calculated X-LA and X-TA (doubly degenerate) frequencies
are shown vs. volume. The transverse modes at the Xpoint decrease with pressure as V is reduced below 0.28V0
(P∼ 106 GPa), but in fact the detailed phonon calculations show that already at that pressure some transverse
mode frequencies for wave vectors along (110) directions
in the Brillouin zone have become imaginary. According
to the calculations Na-fcc becomes dynamically unstable
when the pressure exceeds ∼ 90 GPa. This should be
borne in mind when the results of the Tc calculations in
Fig. 8 are evaluated. This figure shows Tc as calculated
for Na-fcc for three choices of µ∗ , 0.08, 0.10, and 0.13.
The calculations predict that only for pressures close to
the fcc→cI16 transition Na-fcc might be a superconductor, and if so, the transition temperature is low, ∼0.5
K. Below ∼ 90 GPa superconductivity will probably not
be detected. Recalling the trend found for Li, where Tc
reaches a maximum near the phase transformation, it
could be expected that a maximum value near 1 K could
be found also for Na-cI16.
Table III lists the key parameters for the Tc calculations for Na-fcc. The η and λ values are much smaller
than those of fcc-Li, and even rough estimates suggest
that non-zero Tc values cannot be expected for Na, except for the pressure regime where the dynamical instability occurs.
2 0
4 0
6 0
8 0
1 00
1 2 0
P (GPa)
FIG. 8: Tc calculated using linear reponse and the Eliashberg
equations, open circles and curves (just guides-to-the-eye).
The dashed curve is for µ∗ =0.08, dash-dotted for µ∗ =0.10,
and the full-line was obtained with µ∗ =0.13. Note that the
fcc structure of Na becomes dynamically unstable around 90
GPa, and the calculations at 106 GPa were obtained after
omission of some soft phonons (imaginary frequencies) from
the calculations.
TABLE III: Eliashberg results: Na-fcc parameters used in the
Allen-Dynes equation, Eq. 3.
V/V0
0.28
0.4
0.5
0.7
1.0
P (GPa) η (eV/Å2 )
106
0.766
38
0.218
19
0.139
5.0
0.120
0.0
0.128
III.
λ
0.487
0.342
0.069
0.065
0.148
ωlog (K)
130
153
177
190
137
√
< ω 2 > (K)
196
205
222
213
147
DISCUSSION AND CONCLUSIONS
Previous calculations, presented in Ref. 20, predicted
that lithium under pressure becomes a superconductor.
Experiments later confirmed this.22–24 Also, the old calculations, correctly ascribed the increase in Tc with pressure to a softening of the lattice, and not to an increasing
prefactor in the McMillan equation. However, the calculated Tc values were by far too high compared to the experimental results, unless very large values of the MorelAnderson pseudopotential, µ∗ , were assumed. Whereas
the old calculation used the Rigid-Muffin-Tin Approximation, we have now repeated the lithium calculation
using a more refined method, where the phonons and the
electron-phonon interaction are calculated by means of
an ab initio linear-response method37,38 and by solution
of the Eliashberg equations. It appears that results obtained in this way agree much better with experiments,
even if values (0.13) of µ∗ closer to the ”standard values”
7
are applied. It also appears that this kind of calculations,
with the same choice of µ∗ simultaneously apply to Li-fcc
under pressure and to 9R-Li at ambient pressure, where
Tc is a few mK, at most.19
The main reasons for the difference between the old
and the new results can be ascribed to RMTA, which
uses the crude averaging of phonon frequencies (as in
Ref. 20) and neglects the particular effects of strong e-p
coupling in a few (almost) soft phonon states.
It has also been shown that a usual implementation
of the RMTA may well be numerically inaccurate due
the the muffin-tin model (projected densities of states)
itself. This problem is more serious for the alkali metals
than for transition metals in which the d-states are rather
localized in contrast to the long-ranging s- and p states
in Li and Na.
The very recent work, Ref. 29, by Kasinathan et al.
report results of calculations using methods very similar
to ours. They, as well as Maheswara et al.28 , also conclude that the superconductivity in Li originates in the
coupling to the low-frequency modes and the pressureinduced softening. Further, they show29 how the lowest
transverse mode along Γ-K becomes a soft mode at 35
GPa (even lower, it appears), as also found in Ref. 27.
The mesh in the Brillouin zone used in Ref. 29 is very
dense, and this allowed a detailed identification of ”hot
spots” on the Fermi surface with particularly strong ep coupling. Since this work uses the same basic codes
for the Eliashberg calculations as we did (Savrasov and
Savrasov), it may seem surprising that some of the numerical results differ, for example the λ value for small
volumes. For P=20 GPa Kasinathan et al. find a λ equal
to 3.1, whereas we at 26 GPa (i.e. even a bit higher pressure) get λ = 1.37 (Table II). The difference is probably
mainly due to the better sampling in the BZ used in Ref.
29. We only used 29 points in the irreducible wedge. Such
differences become particularly obvious near the pressure
where a phonon mode goes soft. Also the P-V relations
may be somewhat different. In our case pressures were
derived from different full-potential LMTO calculations,
namely those described in Refs. 2 and 4. In our calculations it does not appear that Li-fcc for P=26 GPa
(V/V0 = 0.5) is as close to the dynamical instability as
Fig. 3 in Ref. 29 would suggest for their 20-GPa calculation. At lower pressures we agree much better with
the results obtained by Kasinathan et al. As mentioned
earlier, our calculated transition pressures agree very well
with those observed.2 The pressure range of the fcc phase
is from 7.5 GPa (measured45 ) to 38.3 GPa (measured
at 180 K, Ref. 2). The hR1 phase was found to exist
from 39 to 42.5 GPa, and at higher pressures the cI16
was stable. It would then appear that the prediction of
a dynamical instability in fcc-Li slightly above 20 GPa
cannot agree with the observed structural information.
But the experiments were carried out at high temperatures (180 K), and we do not really have information
about the phase diagram from x-ray diffraction experiments at very low temperatures. The Tc measurements
by Deemyad and Schilling,24 however, provide interesting information, since the small error bars on the data
show that structural transformations at very low temperatures may occur at pressures which differ significantly
from those found for T=180 K. No superconductivity was
observed in lithium below 20 GPa, and this also agrees
with the earlier experiments.22,23 According to theory,
Tc in the fcc phase does not drop abruptly to zero as P
is reduced below 20 GPa. Maybe29 lithium is not in the
fcc phase for P < 20 GPa and T ∼ 5 K.
The present calculations for sodium follow the lines of
those for Li, and also in this case we find that our RMTA
calculations cannot lead to quantitatively reliable results,
the reasons being the same as for Li. The e-p coupling is
much smaller in Na than in Li and fcc sodium is predicted
NOT to be a superconductor, except maybe close to the
upper pressure range of stability of this phase, ≈ 90 GPa.
At this pressure the lattice becomes so soft that Na could
undergo a transition to a superconducting phase, but Tc
would be low, around 1 K. Near 90 GPa we find that
the lowest transverse phonon mode goes soft, and Nafcc is dynamically unstable. This is then similar to the
finding29 in lithium. Also, it is interesting to note that
Na-fcc was experimentally found to melt near 300 K at
a pressure near 100 GPa in Ref. 46. The occurence of
this dynamical instability may explain why the fcc→cI16
transition pressure (130 GPa) obtained from our static
calculations is much higher than the measured value, 103
GPa.
The dynamical instabilities of the fcc lithium and
sodium phases are similar to what was found47,48 for
cesium, showing that the Cs-II→Cs-III transition cannot be isostructural (fcc).49 The fact that a particularly
strong electron coupling to a few phonon states may drive
superconductivity is not a specific feature of alkali metals, but was found50,51 in MgB2 and more recently in
hole-doped diamond.52 Ostanin et al.53 explained the
pressure-induced superconductivity in phosphorous as a
result of phonon mode softening as in the alkali metals.
Although the calculations here are based on solution
of the Eliashberg equations and state-of-the-art ab initio
phonon and e-p coupling calculations, there are still weak
points. One question concerns the phonons: Is the linear
response theory good enough ?- In particular, what happens close to the instabilities with the accuracy of the
”α2 ” calculations ? And then the most unsatisfactory
point, the unknown µ∗ . In our work this is still treated
as an adjustable parameter, and this means that these
calculations do not represent a ”real theory”. In view
of this, it is very important that it has been possible35
to implement the density-functional theory for superconductivity of Ref. 34 in a computer code, and apply this
to various metals, including lithium, and to note that our
conclusions are not invalidated by these new results.
Concerning the quantitative comparison between calculated and measured superconductive properties of the
light alkali metals under pressure, there are still some
”loose ends”. Some questions may be answered if the
8
phase diagrams of the alkali metals were known in more
detail, preferably from x-ray diffraction experiments covering also very low temperatures.
IV.
ACKNOWLEDGMENTS
support from the Danish Natural Science Research Council, grant No. 21-03-0340. The calculations were performed at the Center for Scientific Computing in Aarhus
(CSCAA), financed by the Danish Center for Scientific
Computing (DCSC) and the Faculty of Science, University of Aarhus.
The authors thank S.Y. and D.Y. Savrasov for permission to use their linear-response codes. We acknowledge
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
J.B. Neaton and N.W. Ashcroft, Nature (London) 400,
141 (1999).
M. Hanfland, K. Syassen, N.E. Christensen, and D.L.
Novikov, Nature (London) 408, 174 (2000).
J.B. Neaton and N.W. Ashcroft, Phys. Rev. Lett. 86, 2830
(2001).
N.E. Christensen and D.L. Novikov, Solid State Commun.
119, 477 (2001).
R. Rousseau and D. Marx, Chem. Eur. J. 6, 2982 (2000).
G.G.N. Anginelli, F. Sirengo, and R. Pucci, Eur. Phys. J.
B 32, 323 (2003).
G.J. Ackland and I.R. Macleod, New J. Phys. 6, 138
(2004).
A. Rodriguez-Prieto and A. Bergara, Proc. of the ”Joint
20-th AIRAPT-43-th EHPRG, June 27-July 1, Karlsruhe/Germany 2005; cond-mat/0505619.
R. Rousseau, K. Uehara, D.D. Klug, and J.S. Tse,
ChemPhysChem 6, 1703 (2005).
M. Hanfland, K. Syassen, private communication (2001).
and M. Hanfland, K. Syassen, I. Loa, N.E. Christensen, and D.L. Novikov, Verhandlungen der Deutschen
Physikalischen Gesellschaft (5/2003), 536 (2003).
N.E. Christensen and D.L. Novikov, Ψk Newsletter 42, 76
(2000).
A.Y. Liu and M.L. Cohen, Phys. Rev. B 44 9678 (1991).
T.L. Thorp, B.B. Tripplett, W.D. Brewer, M.L. Cohen,
N.E. Phillips, D.A. Shirley, J.E. Templeton, R.W. Stark,
and P.H. Schmidt, J. Low Temp. Phys. 3, 589 (1970).
W.L. McMillan, Phys. Rev. 167, 331 (1968).
A.Y. Liu, A.A. Quong, J.K. Freericks, E.J. Nicol, and E.C.
Jones, Phys. Rev. B 59, 4028 (1999).
P. Morel and P.W. Anderson, Phys. Rev. 125, 1263 (1962).
G.M. Eliashberg, Sov. Phys. JETP 11, 696 (1960).
J.P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990).
J.K. Freericks, S.P. Rudin, and A.Y. Liu, Physica B 284288, 425 (2000).
N.E. Christensen and D.L. Novikov, Phys. Rev. Lett. 86,
1861 (2001).
G.P. Gaspari and B.L. Gyorffy, Phys. Rev. Lett. 28, 801,
(1972).
K. Shimizu, T. Kimura, S. Furomoto, K. Takeda, K. Kontani, Y. Onuki, and K. Amaya, Nature (London) 412, 3196
(2002).
V.V. Struzhkin, M.I. Eremets, W. Gan, H.K. Mao, and
R.J. Hemley, Science 298, 1213 (2002).
S. Deemyad and J.S. Schilling, Phys. Rev. Lett 91, 167001
(2003).
N.E. Christensen and D.L. Novikov (2002, unpublished); N.E. Christensen; p. 15 in Proceedings of the 4-th Workshop on Computational
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Chemistry and Molecular Spectroscopy, Oct. 2004,
Punta de Tralca, Chile;
ed. R. Arratia-Perez,
http://www.unab.cl/workshop/doc/Proceedings 2004.pdf,
and N.E. Christensen, Verhandlungen der Deutschen
Physikalischen Gesellschaft (5/2003), 534 (2003).
R.E. Alonso, S. Sharma, C. Ambrosch-Draxl, C.O. Rodriguez, and N.E. Christensen, Phys. Rev. B 73, 064101
(2006).
J.S. Tse, Y. Ma, and H.M. Tutuncu, J. Phys.: Condens
Matter 17, S911 (2005).
S.U. Maheswari, H. Nagara, K. Kusakabe, and N. Suzuki,
J. Phys. Soc. Jpn. 74, 3227 (2005).
D. Kasinathan, J. Kunes, A. Lazicki, H. Rosner, C.S. Too,
R.T. Scalettar, and W.E. Pickett, Phys. Rev. Lett. 96,
047004 (2006).
A. Razzaque, A.K.M.A. Islam, F.N. Islam, and M.N. Islam, Solid State Commun. 131, 671 (2004).
K. Iyakutti and C.N. Louis, Phys. Rev. B 70, 132504
(2004).
C.F. Richardson and N.W. Ashcroft, Phys. Rev. B 55,
15130 (1997).
C.F. Richardson and N.W. Ashcroft, Phys. Rev. Lett. 78,
118 (1997).
M. Lüders, M.A.L. Marques, N.N. Lathiokis, A. Floris, G.
Profeta, L. Fast, A. Continenza, S. Massida, and E.K.U.
Gross, Phys. Rev. B 72, 024545 (2005).
G. Profeta, C. Franchini, N.N. Lathiokis, A. Floris, A.
Sanna, M.A.L. Marques, M. Lüders, S. Massida, E.K.U.
Gross, and A. Continenza, Phys. Rev. Lett. 96, 047003
(2006).
O.K. Andersen, Phys. Rev. 12, 3060 (1975).
S.Y. Savrasov, Phys. Rev. B 54, 16470 (1996).
S.Y. Savrasov and D.Y. Savrasov, Phys. Rev. B 54, 16484
(1996).
R.C. Dynes and J.M. Rowell, Phys. Rev. B 11, 1884
(1975).
P.B. Allen and R.C. Dynes, Phys. Rev. B 12, 905 (1975).
T.H. Lin and K.J. Dunn, Phys. Rev. B 33, 807 (1986).
D.G. Pettifor, J. Phys. F 7, 1009 (1977).
H.L. Skriver and I. Mertig, Phys. Rev. B 32, 4431 (1985).
D. Glötzel, D. Rainer, and H.R. Schober, Z. Phys. B 35,
317 (1979).
M. Hanfland, I. Loa, K. Syassen, U. Schwarz, and K. Takemura, Solid State Commun. 112, 123 (1999).
E. Gregoryanz, O. Degtyareva, M. Somayazulu, R.J. Hemley, and H.-K. Mao, Phys. Rev. Lett. 94, 185502 (2005).
N.E. Christensen, D.J. Boers, J. van Velsen, and D.L.
Novikov, Phys. Rev. B 61, R3764 (2000).
N.E. Christensen, D.J. Boers, J. van Velsen, and D.L.
Novikov, J. Phys. Condens. Matter 12, 3293 (2000).
9
49
50
The Cs-III phase was later by McMahon et al. identified as having an orthorhombic structure (space group
C2221 ) with 84 atoms in the unit cell: M.I. McMahon,
R.J. Nelmes, and S. Recki, Phys. Rev. Lett. 87, 255502
(2001).
J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Antropov,
and L.L. Boyer, Phys. Rev. Lett. 86, 4656 (2001).
51
52
53
Y. Kong, O.V. Dolgov, O. Jepsen, and O.K. Andersen,
Phys. Rev. B 64, 020501 (2001)
L. Boeri, J. Kortus, and O.K. Andersen, Phys. Rev. Lett.
93, 237002 (2004)
S. Ostanin, V. Trubitsin, J.B. Staunton, and S.Y.
Savrasov, Phys. Rev. Lett. 91, 087002 (2003).