Any model for the effect of hydrogen on C4KHg must not only account for the Tc increase but also be consistent with other relevant experiments. Before conjecturing what the cause of the rise in Tc might be, it is useful to review some of these other experiments.
Most researchers who have worked with the KHg-GIC's have not measured the superconducting transition temperature. This is entirely understandable considering the time-consuming and expensive nature of the 3He experiments. Unfortunately, the lack of Tc characterization in most papers on C4KHg limits their helpfulness for understanding the sample dependence of Tc and the hydrogen-induced enhancement. Unless the type of C4KHg sample used by another group is known, it is hard to know how their data fit into the overall picture.
Ideally, the most helpful type of information on
C4KHg would be measurements of a physical property
(besides Tc) that was different in pink and
unhydrogenated gold samples, but the same in pink and
hydrogenated gold samples. Despite many attempts to find such
a property (described in Section 
), no such measurement is currently known.
One set of data does have special significance for the
interpretation of the hydrogenation experiments: the
measurements of Tc as a function of applied
pressure in the KHg-GIC's performed by DeLong and
collaborators.[56,58,55] Figure
a) shows the superconducting transition as a function of
pressure for a C4KHg specimen which had
Tc 1.3 K and Delta Tc/
Tc 0.31 at room pressure. The dramatic sharpening
of the transition (final Delta Tc/
Tc 2e-2) and shift of Tc to
approximately 1.5 K under the very small applied pressure of
0.8 kbar are remarkably similar to the effect of
hydrogenation. The smallness of an applied pressure of 0.8
kbar can best be appreciated by noting that dTc/dP
in C8KHg is about -6.5× 10 -5
K/bar,[58] so that 0.8
kbar of applied pressure shifts Tc in
C8KHg by only about 50 mK. In both the
hydrogenation and pressure experiments, a minute perturbation
to the sample radically narrows the superconducting
transition and increases Tc.
Figure: Pressure dependence of
Tc in KHg-GIC's. From Ref. [55]. a) Pressure-induced
transition narrowing in C4KHg. Notice that the
application of a small pressure, 0.8 kbar, increases
Tc to 1.5 K, while application of further pressure
decreases Tc at a rate dTc/dP =
-5×10-5 K/bar. b) Monotonic decline of
Tc with pressure in C8KHg.
dTc/dP = -6.5×10-5 K/bar.
The large sensitivity of the superconductivity in C4KHg to small pressures is particularly interesting to contrast with the slow monotonic decrease of Tc with pressure in C8KHg.[58] The slow decline of Tc with pressure that is observed in C8KHg is typical of nearly-free-electron metals.[58] Nearly-free-electron character is a reasonable model for the KHg-GIC's since band structure calculations[215,112] show substantial intercalant s and p character at the Fermi level. Therefore the behavior of C8KHg could be considered conventional, whereas the transition narrowing observed in C4KHg is quite anomalous. At pressures above the initial discontinuity in Tc, the pressure dependence of C4KHg also becomes conventional, with dTc/dP = -5× 10-5 K/bar.[55]
As pointed out by DeLong and Eklund,[58,55] the Tc(P)
experiments on the KHg-GIC's are reminiscent of those on the
transition metal dichalcogenides. In Section
the strong Tc increase with low pressure of
NbSe2 was contrasted with the weak Tc
increase with pressure of NbS2.[217] The large low-pressure
magnitude dTc/dP of NbSe2 is attributed
to the suppression of a CDW. Once the CDW of NbSe2
has been destroyed by pressure, its dTc/dP is
almost the same as that of the non-CDW compound
NbS2.[217] The
CDW material NbSe3 and its non-CDW relative
TaSe3 also show dTc/dP behavior similar
to the NbSe2/NbS2 pair.
DeLong and Eklund have proposed that C4KHg and C8KHg might be such a CDW/non-CDW matched pair.[55] According to this model, the initially broad transition of C4KHg is due to the presence in some of the sample of a CDW state that is gapping part of the Fermi surface. The portion of the sample that supported a CDW would have a depressed Tc, whereas the non-CDW part of the sample would have Tc = 1.5 K. When a small amount of pressure destroys the CDW, the whole material has the intrinsic Tc = 1.5 K transition.
As an alternative theory, DeLong and Eklund[55] proposed that the pressure
drives an ordering transition in C4KHg. They
mention the improvement of long-range order in the
intercalant layers or an improvement in stacking fidelity as
possible ordering transitions. However, there are good
reasons why an order-disorder transition is unlikely to
produce the behavior seen in Figure
a). First, Tc in most superconductors is not
usually as sensitive to crystallographic order as the
hydrogenation and pressure data would suggest. There are many
examples of disordered superconductors with Tc's
close to those of single crystals. Other superconducting
properties such as the critical fields and currents are much
more sensitive to crystalline order than Tc.
(These transport properties depend directly on the
mean-free-path, as discussed in Section ).
The other reason that the disorder-order hypothesis seems
unlikely is much more fundamental. This second argument,
which is based on thermodynamic considerations, is due to
Clarke and Uher.[43] The
Clarke-Uher argument is based on the observation that the
change of shape of the superconducting transition is almost
entirely reversible. As Figure
shows, when applied pressure is released from a
C4KHg specimen that had been pressurized up to 8
kbar, the sample's superconducting transition returns almost
exactly to its original shape. The small amount of deviation
between the original and final room-pressure transitions can
probably be attributed to plastic deformation of the sample.
If the effect of pressure were genuinely to force a
disorder-order transition, then one would expect the material
to remain ordered when the pressure is removed. The reason is
that the part of the Helmholtz free energy (F = E -
TS) which favors the formation of a disordered phase is the
entropic term. The entropic contribution to the free energy
is proportional to temperature, which is the formal
justification for the observation that higher temperatures
encourage the formation of disordered phases. Metastable
disordered phases do exist at low temperatures since their
rate of transformation is suppressed due to the lack of
thermal energy. However, should disordered material be
transformed to ordered material at low temperature by the
application of pressure, basic thermodynamics suggests that
the materials should remain ordered since the entropic forces
driving disorder are effectively zero at 1 K.
These thermodynamic considerations suggest that a non-hysteretic low-temperature transformation must be of the order-order rather than disorder-order variety. Order-order transitions can be driven by the energy term in the free-energy, and so do not always require thermally assisted growth. In metallurgy a transformation which is not thermally assisted is called displacive or martensitic.[41] One type of martensitic transition is the charge-density wave transition already discussed in connection with the TMDC experiments.[91]
There is additional evidence to support the identification of
the hydrogen- and pressure-induced transformation as an
order-order transition. If the difference between the low-
Tc and Tc = 1.5 K material were merely
the degree of disorder, then it should be impossible to make
a low- Tc sample with a sharp transition. Yet
Table shows that one gold
specimen had Delta Tc/ Tc only
7× 10-2, comparable to the width of the
better Tc = 1.5 K samples. It is true that the
lower Tc samples tend to have broader transitions,
but this is probably just an indication that the lower-
Tc phase is harder to grow. If the beta
phase is metastable, it seems sensible that it would be
harder to grow in a well-ordered condition than the
putatively stable alpha phase.
The importance of the Tc-versus-pressure experiments is therefore twofold. One contribution of the pressure experiments is to reinforce the evidence from the hydrogenation experiments that the low- Tc phase can be destroyed by very small perturbations. The vital contribution of the pressure experiment is that the reversibility of the transformation shows it to be an order-order transition, not the disorder-order transformation also suggested by DeLong and Eklund.[55]
The next logical question is why the low-temperature ordering
should be a CDW transition rather than an ordinary structural
phase transition. C8K has a structural phase
transition near 13 kbar to a Tc = 1.5 K phase, as
discussed in Section .[55,13] The high-pressure phase has a
sqrt3 × sqrt3R30 ° structure.[43] It seems reasonable to
ask whether the low- Tc material in
C4KHg could not also undergo a structural phase
transition to a higher- Tc phase. As remarked in
Section , there is evidence from
the neutron scattering experiments that all lower-
Tc samples contain both the alpha and
beta phases of C4KHg. Therefore one might
well hypothesize that the effect of hydrogen and pressure is
simply to transform the beta phase material into
alpha phase. This hypothesis explains all the
superconductivity data quite well, but it runs afoul of the
neutron diffraction data. In a neutron-diffraction study, Kim
and coworkers found no evidence for a change in the relative
abundance of the alpha and beta phases up to
13.8 kbar.[130] The fraction
of beta phase in the neutron diffraction sample was 2%
both at 1 atm and 13.8 kbar. Kim et el. also saw no
evidence from the diffraction patterns for ordering of the
sample or for changes in the relative intensities of the
peaks up to 13.8 kbar.[130]
This neutron diffraction study effectively rules out any
structural phase transition explanation for the data of
DeLong and Eklund.[55]
Since it appears highly probable that hydrogenation has the
same effect as pressure, the data of Kim and coworkers appear
to rule out a structural phase transition explanation for the
hydrogen experiment as well.
The primary objection to the hypothesis of CDW formation in C4KHg has to be why a CDW has not been observed in any of the aforementioned diffraction experiments. An answer to this question has been put forward by Wilson, DiSalvo, and Mahajan, the discovers of charge-density waves in the TMDC's.[264] After the periodic lattice distortion associated with CDW formation was discovered in TaSe2 using electron diffraction, an intensive search for superlattice lines was made with x-ray and neutron diffraction. Even when researchers knew where these lines were, observing them with x-rays was still quite difficult, requiring special equipment and long exposure times.[265] Wilson et al. explain that the observation of CDW's is made much easier by the dynamical diffraction that occurs with electron beams, but not with photon or neutron beams.[265]
Timp[246] performed extensive
electron microscopy studies on C4KHg samples, but
he never reports any observations below room temperature.
Because CDW's are easily observed only with electron
diffraction[265] or
nowadays, with a scanning tunnelling microscope,[47] there seems to have been no
experimental opportunity to see a possible CDW in
C4KHg. An estimate can be made of the hypothetical
CDW transition temperature in C4KHg using Eqn. .
To calculate TCDW, let the intrinsic Tc
of C4KHg == Tc0 be 1.5 K. Let the
suppressed Tc be 0.8 K. The fraction of the Fermi
surface which is removed by a CDW transition,
N1/N, can be estimated by using the BCS formula
for Tc, assuming the same Debye temperature and
same BCS interaction parameter V for both transition
temperatures. The result is that removal of about 11% of the
Fermi surface will account for the observed Tc
depression in the gold C4KHg phase. Plugging this
fraction into Eqn. gives TCDW =
243 K. This is a very rough estimate for
TCDW since Eqn. is exponentially
dependent on the fraction of FS removed by the CDW
transition. Nonetheless, this estimate gives one hope that a
CDW transition may yet be observed in C4KHg.
Before moving on to further discussion of the CDW hypothesis, it is worth discussing a few alternative ideas. The possibilities of a disorder-order transition or an order-order structural phase transition can be safely eliminated, for reasons discussed above. The optic-phonons explanation of Tc enhancement by hydrogen that is applicable to the transition metals is not promising for C4KHg. The reason that the optic-phonons picture seems unsuitable here is that it cannot explain why after hydrogenation all samples have the same Tc, 1.5 K.
One more suitable possibility might be an explanation based on the electronic properties of C4KHg rather than the lattice modes or structure. Roth and coworkers[207] proposed a model in which a reduction in carrier density due to hydrogenation lowered the Fermi level until it resided in a density-of-states maximum. The idea was that this maximum density-of-states would correspond to a Tc of 1.5 K, and that all subsequent perturbations would move the Fermi level away from the maximum and consequently depress Tc. Roth et al. hypothesized that mercury vacancies might be responsible for a lower Tc in some samples since the Tc of KHg alloys monotonically increases with increasing mercury content.[205]
Since this electronic-structure-based proposal was made,
evidence has accumulated that mercury stoichiometry does not
differ significantly among higher- and lower- Tc
C4KHg samples. This evidence was reviewed in
Section . One might wonder
whether the coexistence of the alpha and beta
phases in the lower- Tc C4KHg specimens
might not shift the Fermi level enough to lower the
density-of-states and suppress Tc. This
possibility is not out of the question, but the similarity of
the pressure and hydrogenation data between the TMDC's and
the GIC's is powerful evidence for the CDW hypothesis.
Furthermore, the very small amount of pressure or hydrogen
needed to increase Tc strongly hints that a phase
transition is involved.
Up to this point there has been no justification for the
claim that C4KHg is the type of solid expected to
undergo a charge-density wave transition. This is no accident
considering that the CDW phenomenon, like superconductivity,
is a subtle collective effect that depends on the details of
band structure, lattice modes, and electron-phonon coupling.
The best argument for a charge-density wave in
C4KHg would seem to be that its Fermi surface, as
calculated by Holzwarth[112] (see Figure
), is quite similar to that of the TMDC's. According to
Holzwarth, a CDW transition in C4KHg seems
possible, but the degree of nesting is very sensitive to the
proposed splitting[70]
of the mercury layers in C4KHg.[113] The similarity of the
C8K Fermi surface to that of TaS2 was
previously noted by Inoshita,[115] who found a possible Fermi
surface nesting wavevector. The hypothetical nesting
wavevector for C8K is shown in Figure .
Figure: Possible Fermi surface nesting wave
vector in C8K. From Ref. [115]. The horizontal
cross-section of the FS in the Gamma-K-M plane is
shown. The arrow indicates the proposed nesting wave vector
near the M point.
If a CDW does occur in C4KHg, it should be visible
in other experiments besides electron diffraction. For
example, a discontinuity in resistivity or susceptibility
might occur. Published resistivity and susceptibility data do
show interesting anomalies, but at different temperatures.[72] The basal-plane
resistivity undergoes a change in slope at about 200 K in
both C4KHg and C4RbHg, as illustrated
in Figure a). The magnetic
susceptibility is flat at 200 K but shows a small anomaly at
about 50 K. Notice that resistivity and susceptibility
anomalies are not observed for the stage 2 specimens. It
would be informative to repeat these measurements, especially
on GIC's whose Tc's have been measured.
Figure: Temperature dependence of the
resistivity and susceptibility in the alkali-metal mercury
GIC's. From Ref. [72].
a) Temperature dependence of the resisitivity. Curves (1) and
(2) are for C4RbHg; (3) is for C4KHg;
(4) is for C8RbHg; and (5) is for
C8KHg. b) Temperature dependence of the
susceptibility. Curves (1) and (2) are for C4KHg;
(3) is for C8RbHg; (4) is for
C4K0.5Rb0.5Hg; (5) is for
C4RbHg, and (6) is for C8KHg.
A summary of the CDW hypothesis for C4KHg is in order. The basic picture is that proposed by DeLong and Eklund.[55] This picture relies heavily on analogies to the much-studied transition metal dichalcogenides. In almost any reasonable model, the main difference between the pink and gold phases of C4KHg is that the gold samples contain the beta phase. Specific to the model described here is the idea that the beta phase is unstable to the formation of a CDW which opens a gap on some of the Fermi surface at a temperature well above Tc. The removal of some of the FS lowers Tc in the gold samples from the intrinsic value of 1.5 K to about 0.8 K. Hydrogenation and pressure suppress the CDW, presumably by destroying the FS nesting. Destruction of the CDW close its gap, and restores Tc to 1.5 K. Removal of pressure allows reformation of the CDW and a return of Tc to 0.8 K. This model is by no means proven, but it explains all the available data and is reasonable because of the similarity of the superconducting GIC's and TMDC's.