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## Physics.georgetown.edu

**Structural phase stability and electron-phonon coupling in lithium**
*Department of Physics, Georgetown University, Washington, DC 20057-0995*
*Lawrence Livermore National Laboratory, Livermore, California 94550*
*Department of Physics, Georgetown University, Washington, DC 20057-0995*
*Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1*
*Department of Physics, Georgetown University, Washington, DC 20057-0995*
First-principles calculations of the free energy of several structural phases of Li are presented. The density-
functional linear-response approach is used to calculate the volume-dependent phonon frequencies needed forcomputing the vibrational free energy within the quasiharmonic approximation. We show that the transforma-tion from a close-packed structure at low temperatures to the bcc phase upon heating is driven by the largevibrational entropy associated with low-energy phonon modes in bcc Li. In addition, we find that the strengthof the electron-phonon interaction in Li is strongly dependent on crystal structure. The coupling strength issignificantly reduced in the low-temperature close-packed phases as compared to the bcc phase, and is con-sistent with the observed lack of a superconducting transition in Li. ͓S0163-1829͑99͒11905-2͔

**I. INTRODUCTION**
Neutron experiments3,8,9 show that while there may be
The pressure-temperature structural phase diagram of
some softening of certain phonon modes in bcc Li upon
solid Li, arguably the simplest metal, is not well established.

cooling, the transformation occurs well before any phonon
Even along the zero-pressure axis, Li exhibits some complex
frequencies or elastic constants approach zero, ruling out the
behavior. At ambient pressure and room temperature, Li
soft-mode mechanism10–12 for the transformation. The trans-
crystallizes in the bcc structure, but upon cooling, it under-
formation more likely results from a competition between
goes a martensitic transformation around 80 K. The transfor-
the internal energies and entropies of the different phases.

mation was first observed in 1956,1 but the crystal structure
For example, Friedel13 argued on general grounds that the
of the low-temperature phase remained a subject of debate
phonon spectrum should scale roughly with coordination
for several decades. In 1984, Overhauser suggested that the
number, leading to an overall lowering of the bcc phonon
neutron scattering data were consistent with the 9R
spectrum compared to that of close-packed structures. The
structure,2 a close-packed phase with a nine-layer stacking
resulting larger vibrational entropy associated with the bcc
sequence. Subsequent investigations confirmed 9R as the pri-
structure provides a qualitative explanation for the preva-
mary structure at low temperatures,3 but also showed that it
lence of the bcc structure as the high-temperature phase in
is accompanied by a large and variable amount of a disor-
metals. While these ideas are believed to apply to many sys-
dered polytype consisting of short-ranged fcc- and hcp-type
tems that exhibit martensitic transformations, it is useful to
stacking order, as well as some amount of bcc.4 Furthermore,
examine them in the context of detailed materials-specific
upon heating, the 9R phase and disordered polytype appear
phonon spectra which are accessible from first-principles cal-
to transform first to an ordered fcc phase before reverting to
Several first-principles theoretical investigations have fo-
Little information is available about the phase diagram of
cused on the relative stability of different crystal structures
Li at higher pressures. Upon compression, the temperature at
for Li at zero pressure.14,15 Conflicting results have emerged,
which the martensitic transformation begins,

*M s *, rises at a
but all the studies agree that the energy differences between
rate of about 20 K/GPa, at least up to pressures of 3 GPa,5,6
structures are small ͑much less than 1 mRy/atom͒, making it
with the structural characteristics of the low-temperature
difficult to determine the ordering reliably. Most of the the-
phase remaining the same as at ambient pressure. At room
oretical work has been based on zero-temperature calcula-
temperature, a structural transition is observed at

*P*
tions that do not even include zero-point energies. Given that
ϭ6.9 GPa.7 The crystal structure of the compressed phase
Li is such a light atom, finite-temperature effects arising
was initially identified as fcc, but the limited diffraction data
from vibrational degrees of freedom are likely to play a role
STRUCTURAL PHASE STABILITY AND ELECTRON- . . .

in the structural energetics of this system. In a recent study,
cal density approximation ͑LDA͒. The Li pseudopotential is
vibrational contributions to the free energies of bcc and fcc
generated by the Troullier-Martins method.28 The Wigner
Li were computed within the harmonic approximation.15 The
interpolation formula is used for the correlation potential,29
fcc phase was found to be favored at

*T*ϭ0 K, which is
and a partial core correction is used to treat the nonlinearity
consistent with experiments. Furthermore, the free-energy
of the exchange and correlation interaction between the core
difference between the two phases was calculated to decrease
and valence charge densities.30 Since the structures lie very
with increasing temperature, but, in contrast to what is ob-
close in energy, large sets of

**k **points are needed to reliably

served in experiments, the fcc phase was calculated to re-
compare the total energies of different structures. We use
main thermodynamically stable up to at least a few hundred
1120, 1122, 1300, and 918 points in the bcc, 9R, fcc, and hcp
irreducible Brillouin zones ͑IBZ͒, respectively, to obtain to-
In this paper, we go beyond the purely harmonic approxi-
mation and include anharmonic effects at the quasiharmonic
10Ϫ5 Ry/atom with respect to BZ sampling. In addition, a
level.16 This approximation, in which the volume depen-
large plane-wave cutoff of 20 Ry is used in the calculation of
dence of the phonon spectrum is explicitly included, hasbeen shown to provide a remarkably good description of the
temperature dependence of the lattice constant and bulk
The free energy at finite temperature has contributions
modulus of bcc Li up to its melting point.17 The density-
from both electronic excitations as well as lattice vibrations.

functional linear response method18,19 is used to compute the
While the effects of thermal excitation of electrons are ex-
volume-dependent phonon frequencies. We find that a quasi-
pected to be weak at the temperatures of interest, their con-
harmonic treatment of the phonon free energies reproduces
tributions to the free energy can easily be included by using
the experimentally observed bcc-9R transformation, though
the transition temperature is overestimated. The thermody-
standard DFT total energy.31 The lattice excitations are
namic stabilization of the bcc structure at high temperatures
treated within the quasiharmonic approximation,16 where the
is shown to be completely attributable to its excess phonon
full Hamiltonian at a given volume is approximated by a
harmonic expansion about the equilibrium atomic positions,
A possibly related issue is the lack of a superconducting
but anharmonic effects are included through the explicit vol-
transition in Li. Specific heat measurements,20 as well as
ume dependence of the vibrational frequencies. The Helm-
first-principles calculations21–25 suggest that the electron-
phonon mass enhancement parameter in Li is Ϸ0.45. Usingthe McMillan equation,26 with a Coulomb repulsion param-
ͫ ͩប

**q**͑

*V*͒ͪͬ

eter of *ϭ0.12, which is believed to be typical for simple

*F*͑

*V*,

*T*͒ϭ

*F *͑
el

*V *,

*T *͒ ϩ

*k BT *͚ ln 2 sinh
metals, we would expect to find a superconducting transition
in Li at temperatures on the order of 1 K. Yet no transition
where

**q**(

*V*) is the frequency of the th phonon band at the

has been observed in experiments, at least down to 6 mK.27
point

**q **in the Brillouin zone.

Because of uncertainties in the low-temperature crystal
The density-functional linear-response method is used to
structure, most of the previous calculations, as well as the
compute the phonon spectra for each structure at several dif-
analysis of the specific heat data, were based on bcc Li. One
ferent volumes. The dynamical matrices are computed using
calculation of the electron-phonon coupling strength in 9R Li
the scheme described in Ref. 19, where the self-consistent
suggested that the coupling is indeed weaker in the low-
change in the Hamiltonian caused by ionic displacements is
temperature phase.24 However, there were large uncertainties
obtained by solving a Bethe-Saltpeter equation for the
in that result since it was based on frozen-phonon calcula-
change in the charge density. At each volume, the dynamical
tions in which it was only feasible to examine the coupling
matrix is computed for 285, 240, 98, and 95 phonon wave
constants at a few phonon wave vectors. Using the linear-
vectors in the bcc, fcc, 9R, and hcp IBZs, respectively. A
response approach rather than the frozen-phonon method, we
planewave cutoff of 12 Ry is used in the calculation of pho-
are able to efficiently sample wave vectors throughout theBrillouin zone, and more accurately calculate the structure
dependence of the electron-phonon interaction in Li in thiswork. Our results show that in 9R, fcc, and hcp Li, is

**B. Results and discussion**
significantly smaller than in bcc Li. We argue that the calcu-
The equilibrium volumes and energies calculated for bcc,
lated coupling in the close-packed phases is weak enough to
fcc, hcp, and 9R Li are listed in Table I. For volumes within
be consistent with current experimental limits on the transi-
about Ϯ12% of the equilibrium volume, we find that the
stacking of the close-packed planes in 9R Li is near ideal:
The remainder of this paper is organized as follows. In
when structural parameters are chosen to correspond to ideal
Sec. II, we outline the methods used to compute the freeenergies and present our results for the structural phase sta-
stacking, the calculated forces on the atoms are less than
bility of Li. Section III focuses on the comparison of the
10Ϫ4 Ry/au and the differences between diagonal compo-
electron-phonon interaction in bcc and 9R Li. A summary
nents of the stress tensor are less than 2ϫ10Ϫ6 Ry/au3.

and concluding remarks are given in Sec. IV.

This is consistent with neutron data which show the

*c*/

*a *ratioto be within 0.1% of the ideal value.3 Similarly, for the hcp

**II. STRUCTURAL PHASE STABILITY OF LITHIUM**
structure, the optimal value for

*c*/

*a *is found to be 1.63,which is very close to the ideal close-packed value of 1.633.

**A. Method**
The

*T*ϭ0 K static calculations, which include only the
The zero-temperature structural energetics are calculated
LDA total energies, show bcc Li to be the least favorable of
using the plane-wave pseudopotential method within the lo-
all the structures considered. Bcc Li lies about 0.1 mRy/atom

TABLE I. Equilibrium volume and energy calculated for bcc,
fcc, 9R, and hcp Li. Results are presented for both static zero-temperature LDA calculations, as well as zero-temperature calcula-tions that include zero-point energies. For hcp Li, only static resultsare presented because full quasiharmonic calculations were not per-formed for this phase.

above the close-packed structures, all three of which havenearly the same energy. We find the lowest lying structure tobe fcc, with 9R and hcp Li only about 0.01 mRy/atom higher
FIG. 1. Calculated Helmholtz free energy versus volume for bcc
and 9R Li at two temperatures. The curves are fits of the calculatedpoints to the Birch equation of state.

The relative ordering of various structures for Li has been
the subject of some controversy. Several static LDA studieshave been carried out, with varying results.14,15 Because the
served experimentally. Interestingly, this close-packed to bcc
different calculations differ in the basis sets employed, the
transition was not found in a recent LDA calculation based
sampling of the BZ, the form of the LDA potential, and the
on the purely harmonic approximation.15 In that work, free
shape approximations used, it is difficult to sort out the rea-
energies for bcc and fcc Li were compared, and fcc Li was
sons for the discrepancies. Nevertheless, all the calculations
found to be energetically favored all the way from

*T*ϭ0 K
agree that the structures are close in energy, and hence are at
up to several hundred kelvin. In the present work, we find
least qualitatively consistent with the coexistence of various
that the temperature range over which the close-packed
phases at low temperatures. Given the structural similarity
phase is favored indeed expands if the volume dependence of
between the various close-packed phases, their near degen-
the phonon spectrum is neglected, but we still find the bcc
eracy in this monovalent

*s p *metal is not surprising. While
phase to be favored over both fcc and 9R Li at 300 K.

the calculated total-energy differences between the close-
To pin down the transition temperature, it is useful to
packed phases are too small to allow us to definitively deter-
examine the Gibbs free energy at zero pressure. The differ-
mine their relative ordering, we are confident in our result
ence between the 9R and bcc zero-pressure Gibbs energies,
that the close-packed phases are clustered close in energy,
⌬

*G*tot(

*P*ϭ0,

*T*), is plotted as a function of temperature in
and that all are more stable than the bcc phase.

Fig. 2. The calculation indicates that 9R Li is thermodynami-
The average phonon frequency is calculated to be about
cally stable below about

*T*ϭ200 K. Based on the uncertain-
1% lower in bcc Li than in the close-packed phases. Thus
ties in the calculated energy differences, we estimate error
when zero-point energies are included, the energy difference
bars of about Ϯ50 K on the calculated transition tempera-
between bcc Li and the close-packed phases decreases. Wefind however that the relative ordering of the structures re-mains the same. As can be seen in Table I, the inclusion ofzero-point energy also leads to an expansion in the equilib-rium volume. For bcc Li, the static LDA energies underesti-mate the volume by nearly 7%. Since the atoms are so light,vibrational effects are significant, and when zero point mo-tion is taken into account, the error in the calculated volumebecomes about 4.5%, which is more typical of the size oferrors found in LDA studies of other simple metals.

Since the close-packed phases are virtually indistinguish-
able energetically, we choose the 9R structure, which is thepredominant low-temperature phase, as representative of allthe close-packed structures. For the remainder of this section,we focus on comparing the energetics of bcc and 9R Li.

The Helmholtz free energy for bcc and 9R Li are plotted
as a function of volume in Fig. 1. Results are presented for

*T*ϭ0 K and

*T*ϭ300 K. For each crystal structure, calcula-
FIG. 2. Calculated difference in Gibbs free energy between 9R
tions are carried out for six to eight volumes, and the results
and bcc Li at zero pressure, plotted as a function of temperature.

are fit to the Birch equation of state.32 With increasing tem-
The triangles show the difference in electronic contributions to the
perature, the ordering of the phases reverses, and the bcc
free energy and the squares show the difference in phonon contri-
phase is calculated to be stable at room temperature, as ob-

STRUCTURAL PHASE STABILITY AND ELECTRON- . . .

TABLE II. Electronic and electron-phonon-related properties
computed for bcc, fcc, 9R, and hcp Li: the electronic density ofstates at the Fermi level

*N*(

*EF*), a logarithmic average of the pho-non frequencies log , the electron-phonon mass enhancement pa-rameter , and the minimum value of the Coulomb pseudopoten-tial, * , that is consistent with the observed lack of a
superconducting transition above 6 mK.

ture. Since the transformation actually takes place via nucle-ation and growth of the close-packed phase within the bccmatrix, the experimentally observed starting temperature forthe transition upon cooling, about 80 K, provides a lower
FIG. 3. ͑a͒ Phonon density of states ͑per atom͒ and ͑b͒ electron-
bound on the thermodynamic transition temperature. Simi-
phonon spectral function calculated for bcc and 9R Li.

larly, the experimental starting temperature for the close-packed to bcc transformation upon heating, about 150 K,
pletely by the excess vibrational entropy arising from the
low-energy shear modes in the bcc phase.

The triangles in Fig. 2 show the 9R-bcc difference in the
The Clausius-Clapeyron equation allows us to estimate
electronic contributions to the free energy, ⌬

*G*
the slope of the phase line from the changes in entropy and
volume at the transition:

*dT*/

*d P*ϭ⌬

*V*/⌬

*S*ϭ23 K/GPa at

*T*⌬

*S*el . This quantity shows negligible temperature
dependence and favors the close-packed 9R phase through-

*P*ϭ0 GPa. This is in good agreement with the experimen-
out the range of temperatures considered. In fact, ⌬

*G*
tally determined values for the pressure dependence of the
dominated by the difference in internal energies, ⌬

*E*
onset temperature, which range from

*d M s */

*dT*ϭ15 K/GPa
which is over an order of magnitude larger than

*T*⌬

*S*
to 30 K/GPa.5,6 In principle, the entire phase line can be
ϭ300 K. At low temperatures, the leading term in the elec-
determined by analyzing the free energies at different pres-
tronic entropy is proportional to the electronic density of
sures. However, as noted in Refs. 34 and 15, the LDA pre-
states at the Fermi level

*N*(

*E*
dicts a negative Gruneisen parameter for the

*T*1
monovalent metal, the electronic density of states depends
in bcc Li, and an elastic instability at volumes corresponding
weakly on crystal structure ͑Table II͒, and hence the elec-
to pressures of

*P*Ͼ2.5 GPa at

*T*ϭ0 K ͑or

*P*Ͼ3.1 GPa at
tronic entropy plays a negligible role in the structural transi-

*T*ϭ300 K). Experimentally, bcc Li indeed becomes un-
stable upon compression, and transforms to a close-packed
The squares in Fig. 2 show the difference between the 9R
phase, but the transformation occurs at a higher pressure of
and bcc phonon free energies. The difference in vibrational
about 7 GPa. The underestimation of the transition pressure
energies between bcc and 9R Li is virtually temperature in-
indicates that anharmonic effects beyond the quasiharmonic
dependent, at least up to

*T*ϭ400 K, and it is the entropic
level become increasingly important with compression, and a
term in the free energy that stabilizes the bcc structure with
treatment that includes phonon-phonon interactions that can
increasing temperature. The phonon densities of states,

*F*(), for the two phases are plotted in Fig. 3͑a͒. While themaximum phonon frequency is nearly the same in the two

**III. ELECTRON-PHONON COUPLING STRENGTH**
structures, the bcc phonon spectrum has more weight in the

**IN LITHIUM**
low-frequency regime ͑below about 15 meV͒. This enhanced

**A. Method**
weight arises primarily from the

*T *͓

*¯ *0͔ polarization, and surrounding re-
The matrix element for scattering of an electron from
gions in the BZ. The entire

*T *͓
state

*n***k **to state

*n*Ј

**k**Ј by a phonon with frequency

**q** and

meV, and none of the close-packed structures have compa-
eigenvector ⑀

*ˆ***q** is given by

rable low-energy phonon branches. These low-energy shearmodes in bcc Li are easy to excite and give large contribu-
tions to the vibrational entropy. Over the range of tempera-

*g*͑

*n***k**,

*n*Ј

**k**Ј,

**q**͒ϭͱ ប ͗

*n***k**͉⑀

*ˆ*
*n*Ј

**k**Ј͘, ͑2͒

**q**• ␦

*V *SCF

tures considered, the vibrational entropy arising from phononmodes with frequency smaller than 8.5 meV is about twice
where ␦

*V*SCF is the self-consistent change in the potential
as large in bcc Li than in 9R Li, while the entropy associated
due to atomic displacements. This scattering gives rise to a
with higher-frequency modes is about the same in the two
structures. From the thermodynamic standpoint, the close-packed to bcc transformation in Li is therefore driven com-

**q**ϭ 2

**q**͓

*N*͑

*E F*
*g***q**͉2͘͘,

where the double brackets ͗͗ ͘͘ denote a doubly constrainedFermi surface average as defined in Ref. 36. This scatteringprocess also contributes to the effective mass of the electronsvia the mass enhancement parameter
The total mass enhancement parameter, , is obtained bysumming over branches and averaging over wave vectorsthroughout the Brillouin zone.

Within the Eliashberg theory of superconductivity, the
electron-phonon spectral function, which measures the effec-tiveness of phonons of a given energy to scatter electrons onthe Fermi surface, plays a central role. The spectral functionis given by
FIG. 4. ͑a͒ Phonon linewidths and ͑b͒ electron-phonon coupling

**q** ͒ប

parameters calculated for bcc Li along high-symmetry directions in
the Brillouin zone. The coupling parameters

**q** are weighted by a

Within this framework, the mass enhancement parameter is
proportional to the inverse-frequency moment of the spectral
note longitudinal branches, while open symbols indicate transverse
branches. Along the ͓110͔ direction (⌫ to

*N*), the open squares
In the linear-response approach, ␦

*V*SCF is computed in the
correspond to the

*T*1 branch with polarization along ͓11
process of determining the dynamical matrices. Therefore we
open circles correspond to the

*T*2 branch with polarization along
calculate the matrix elements on the same grid of phonon
͓001͔. The dashed curve ͑shown only along the ͓100͔ direction͒
wave vectors used in the phonon calculations described in
corresponds to linewidths calculated for a free-electron model.

the previous section. The doubly constrained Fermi surfaceaverage of

*g *is computing using Bloch functions at 728 ͑650͒
density of states, the bcc phase has enhanced spectral weight

**k **points in the bcc ͑9R͒ IBZ, with delta functions at the

at low frequencies compared to the close-packed structures.

Fermi level replaced by Gaussians of width 0.02 Ry. All of
Since is proportional to the inverse-frequency moment of
our studies of the electron-phonon interaction are carried out

*F*, this difference at low frequencies leads to significant
at the experimental volume for bcc Li at 78 K since previous
differences in . As can be seen in Table II, we find
work has shown that this yields phonon frequencies that
ϭ0.45 for bcc Li, which is in good agreement with earlier
agree well with measured dispersion curves.25 Furthermore,
first-principles calculations,21–25 and smaller values ranging
extrapolations based on available thermal expansion data
from 0.33 to 0.37 for the three close-packed structures.

suggest the equilibrium volume changes by less than 0.5%
Further evidence for the importance of contributions
between

*T*ϭ0 K and

*T*ϭ78 K.37
from low-frequency modes can be found in the values
The calculated ␣2

*F *functions are used as input to the
of log tabulated in Table II. A logarithmic moment of
Eliashberg equations. By solving the Eliashberg equations,
␣2

*F*, this parameter can be expressed as log
we obtain the superconducting transition temperature

*T*
ϭexp͓͚

**q**

**q**log(

**q**)/͚

**q**

**q**͔. Although the average pho-

function of the Coulomb pseudopotential *. The Eliash-
non frequencies for the different structures differ by only
berg equations are solved numerically on the imaginary-
about 1%, log is nearly 30% smaller in bcc Li than in the
close-packed phases. This indicates that some low-frequency
the maximum phonon frequency. The repulsive Coulomb pa-
phonon modes in bcc Li couple strongly to the electrons,
rameter * is a function of the cutoff frequency, and the
thereby dramatically lowering the -weighted logarithmic
value used in the numerical solution of the Eliashberg equa-
tions is not necessarily what should be used in approximate
The wave-vector- and branch-dependent phonon line-
widths and coupling parameters for bcc Li are plotted along

*c *equations such as McMillan’s.26 In fact, it is not com-
pletely clear which value of * is most appropriate for such
high-symmetry directions in Fig. 4͑a͒. The dashed curve
along the ͓100͔ direction shows the linewidths calculated for
For the results reported here, * is scaled to ϭ
a free electron gas with the same average electron density as
bcc Li. Within the simple free-electron model, which as-
1/*(max)ϭ1/*(

*c*)ϩln(

*c */max)
sumes a spherical Fermi surface, Thomas Fermi screening,
and a single plane wave for the wave function, umklappprocesses are neglected and only longitudinal phonon modes

**B. Results and discussion**
couple to electrons. The model yields a linear

*q *dependence
The computed electron-phonon spectral functions for bcc
for ␥ at small wave vectors and gives a reasonable descrip-
and 9R Li are plotted in Fig. 3͑b͒. The fcc and hcp ␣2

*F*
tion of the calculated linewidths for longitudinal modes near
curves resemble the 9R results and are omitted for clarity.

the zone center. However, the full calculation finds sizable
For each structure, the spectral function is similar in shape to
linewidths for transverse modes along all the directions plot-
the corresponding phonon density of states, and as with the

STRUCTURAL PHASE STABILITY AND ELECTRON- . . .

FIG. 5. ͑a͒ Phonon linewidths and ͑b͒ electron-phonon coupling
FIG. 6. Superconducting transition temperature versus * for
parameters calculated for bcc Na along high-symmetry directions in
bcc and 9R Li. The experimental upper limit on

*T*
the Brillouin zone. The coupling parameters

**q** are weighted by a

modes along ͓0͔ which contribute so strongly to in Li.

note longitudinal branches, while open symbols indicate transverse
Thus despite similarities in their electronic structure and
branches. Along the ͓110͔ direction (⌫ to

*N*), the open squares
phonon spectra, bcc Na and bcc Li differ significantly in
correspond to the

*T*1 branch with polarization along ͓11
their electron-phonon coupling parameters.

open circles correspond to the

*T*2 branch with polarization along
The superconducting transition temperatures calculated
͓001͔. The dashed curve ͑shown only along the ͓100͔ direction͒
for bcc and 9R Li are plotted as functions of *(max) in
corresponds to linewidths calculated for a free-electron model.

Fig. 6. For bcc Li, the standard value of *ϭ0.12 yields

*T *ϭ
0.8 K, similar to earlier results in which estimates for
Figure 4͑b͒ shows that the low-lying

*T *͓
were input into the McMillan equation. As can be seen in
modes give anomalously large contributions to . Yet Fig.

Fig. 6, a value of *ϭ0.27 is needed in order to suppress

*Tc*
4͑a͒ shows that the linewidths for these modes are compa-
below the experimental limit of 6 mK. For 9R Li, the tran-
rable in magnitude to those for transverse modes along other
sition temperature is found to be consistent with the experi-

**q** is proportional to ␥

**q** /

mental limit if *ϭ0.19. An approximate upper limit on *
does not depend explicitly on

**q** , the large values of

**q**

can be obtained by letting become infinite in the two-
square well expression *(max)ϭ/͓1ϩ ln(

*EF */max)͔.

small phonon frequencies rather than from Fermi surface
For all the structures considered here, this condition leads to
nesting or other properties related to the electronic structure.

*(max)Ͻ0.23. This confirms that the large value needed
It is interesting to compare these results for bcc Li to
to suppress

*Tc *below the experimental limit in bcc Li is most
results for bcc Na, which also has a low-lying

*T *͓
likely unphysical. For the 9R structure, as well as the other
non branch. The phonon dispersion curves of bcc Na are
close-packed structures, the electron-phonon interaction is
similar to those of bcc Li, with the frequencies scaled by the
weak enough to be consistent with the lack of a transition
square root of the ratio of atomic masses.39,40 Yet first-
above 6 mK, provided that the Coulomb pseudopotential is
principles calculations40,41 for bcc Na find Ϸ0.2, which is
somewhat larger than is commonly assumed. A realistic cal-
less than half the value calculated for bcc Li. Since the de-
culation of * is extremely difficult since it requires know-
gree of Fermi surface nesting, as measured by ͗͗1͘͘, and the
ing the full frequency- and momentum-dependent dielectric
electronic density of states,

*N*(

*EF*), are similar in the two
response function of the system. In a recent paper, static
systems, the differences in electron-phonon coupling arise
momentum-dependent dielectric matrices calculated within
from differences in the matrix elements. There is a particu-
the LDA were used to estimate the Coulomb repulsion pa-
larly striking difference in the phonon linewidths along the
rameters in bcc and 9R Li.42 The estimates of *Ϸ0.16 for
͓0͔ direction, as can be seen by comparing Figs. 4 and 5.

both bcc and 9R Li indicate that the standard values assumed
For both of the transverse branches along this direction, we
for * are not appropriate for this material. This is also

**q**

*M *͔ Li / ͓ ␥

**q**

*M *͔ Na

supported by a recent study of solutions to the Eliashberg
the deeper

*p *pseudopotential in Li arising from the lack of

*p*
equation for the electron gas which found that the role of
states in the Li core. The stronger potential gives rise to a
Coulomb repulsion is significantly underestimated in low-
larger change in the bare potential ␦

*V*bare when atoms are
displaced. Furthermore, because the

*p *orbitals are more
The thermal effective mass derived from heat capacity
tightly bound in Li, the electrons are less effective in screen-
measurements provides additional information on the
ing ␦

*V*bare . In combination, these two effects result in sig-
electron-phonon mass enhancement parameter. The thermal
nificantly larger electron-phonon matrix elements in Li than
mass is related to the band mass according to

*m *ϭ
in Na. In Na the electrons are able to very effectively screen
ϩ) and is found experimentally from the electronic contri-
the change in ionic potential, particularly for the transverse
bution to the specific heat. Measurements for Li yield

*m*th
ϭ2.22

*m*, where

*m *is the bare electron mass.20 The band
yielding a transition temperature within about 100 K of the
mass for 9R Li is calculated to be about 7% larger than that
observed starting temperature

*M s *.

for bcc Li, which offsets the difference in . The calculated
The thermodynamic properties addressed in this work
provide a starting point for understanding the details of the
2.25

*m *and 2.24

*m *for bcc and 9R Li,
respectively, are both in good agreement with the specific
structural transformation in Li. Many interesting questions
heat data. Note however that this analysis does not place
remain about the kinetics of the transition, including for ex-ample, details about the reaction path,44 the role played by
strict constraints on since contributions to the mass renor-
coherency stresses in determining which close-packed phase
malization arising from electron-electron manybody effects
appears,4 the appearance of the fcc phase intermediate be-
beyond the LDA have been assumed to be negligible.

tween 9R and bcc upon heating,4 and the importance of het-
Our results for the electron-phonon interaction in 9R Li
are thus consistent both with the observed lack of a super-
conducting transition as well as with the specific heat data.

0 ͔ phonon modes which are responsible for
One complication is that since Li is never observed to trans-
stabilizing the bcc phase at high temperatures also play an
form completely to a perfect 9R lattice—there is always a
important role in the variation of the electron-phonon cou-
mixture of the disordered close-packed polytype, as well as
pling strength with crystal structure. It is primarily contribu-
some amount of bcc, the exact amounts of which depend on
tions from these modes that make the coupling parameter
thermal history—the exact structural characteristics of the
about 30% larger in bcc Li than in the close-packed phases.

samples used in the various experiments are not known. De-
The coupling in the 9R structure, which is the predominant
pending on the details of how grains with different crystal
phase at low temperatures, is weak enough to be consistent
structures are distributed within the samples, the structural
with the current experimental upper limit on

*Tc *, provided
disorder itself may play a role in the suppression of super-
that a value of *(max)Ϸ0.19 is assumed. This is a plau-
conductivity in this material. We hope this work motivates
sible value given recent theoretical work that shows that the
future experimental studies on well-characterized samples of
Coulomb repulsion parameter in low-density metals like Li
low-temperature Li and on possible connections between the
can significantly exceed the standard values assumed.42,43
microstructure of the samples and electronic and transport
The calculated enhancement of the electronic mass due to the
electron-phonon interaction in 9R Li is also consistent withavailable specific heat data.

**IV. CONCLUSIONS**
**ACKNOWLEDGMENTS**
We have used the first-principles linear-response method
to compute the vibrational properties of Li in various crystal
We thank D.W. Hess and B. Sadigh for helpful discus-
structures. The low-energy

*T *͓
sions. This work was supported by National Science Foun-
correspond to shearing of ͑110͒ planes, distinguish the vibra-
dation Grant No. DMR9627778 and by the United States
tional properties of the bcc phase from those of the close-
Department of Energy, Office of Basic Science, Division of
packed phases. In particular, while the close-packed 9R, fcc,
Materials Science. A.Y.L. acknowledges the support of the
and hcp phases are energetically favorable at

*T*ϭ0, the large
Clare Boothe Luce Fund. E.J.N. was supported by the Re-
phonon entropy associated with the low-energy modes in bcc
search Corporation and by the Natural Sciences and Engi-
Li stabilizes the bcc structure upon heating. The quasihar-
neering Research Council of Canada. Supercomputing time
monic approximation gives a reasonable description of the
was provided by the NSF at the Pittsburgh Supercomputing
thermodynamics of the transformation at zero pressure,
Center and the San Diego Supercomputer Center.

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Source: http://www.physics.georgetown.edu/~jkf/publications/lithium_prb_99.pdf

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