Table Of ContentPinning frequencies of the collective modes in α-uranium
B. Mihaila, C. P. Opeil, F. R. Drymiotis, J. L. Smith, J. C. Cooley, M. E. Manley, A. Migliori,
C. Mielke, T. Lookman, A. Saxena, A. R. Bishop, K. B. Blagoev, D. J. Thoma, and J. C. Lashley
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
B.E. Lang, J. Boerio-Goates, and B.F. Woodfield
6
Department of Chemistry and Biochemistry, Brigham Young Universtiy, Provo, Utah 84602, USA
0
0
2 G.M. Schmiedeshoff
Department of Physics, Occidental College, Los Angeles, California 90041, USA
n
a
Uraniumistheonlyknownelementthatfeaturesacharge-densitywave(CDW)andsuperconduc-
J
tivity. Wereport a comparison of the specific heat of single-crystal and polycrystalline α-uranium.
2
Away from the the phase transition the specific heat of the polycrystal is larger than that of the
2
single crystal, and the aim of this paper is to explain this difference. In the single crystal we find
] excesscontributionstotheheatcapacityat41K,38K,and23K,withaDebyetemperature,ΘD =
r 256K.Inthepolycrystallinesampletheheatcapacitycurveisthermallybroadened(ΘD =184K),
e butnoexcessheatcapacitywasobserved. Theexcessheatcapacity,C (takenasthedifferencebe-
φ
h
tweenthesinglecrystalandpolycrystalheatcapacities)iswelldescribedintermsofcollective-mode
t
o excitationsabovetheirrespectivepinningfrequencies. Thisattributionisrepresentedbyamodified
. Debyespectrum with two cutoff frequencies, a pinning frequency, νo, for the pinned CDW (due to
t
a grainboundariesinthepolycrystal),andanormalDebyeacousticfrequencyoccurringinthesingle
m crystal. We explain the 50-year-old difference in Debye temperatures between heat capacity and
ultrasonic measurements.
-
d
n PACSnumbers: 71.45.Lr,74.40-s,05.70.Fh,05.70.Jk
o
c
[ New ground states and modulated structures arise in vates the collective mode to finite frequencies expressed
low-dimensional solids [1], but are of particular interest by a threshold electric field, E . The threshold field is
1 th
when they occur in solids that are not low-dimensional, usefully characterized within a single-particle model [4].
v
9 such as α-uranium (α-U) [2] . When periodic, these or- In this model, the collective mode is described by the
9 dered structures may be a spin-density wave (SDW) as equationofmotionofa classicalparticlemovingina pe-
4 found in chromium [3] or a charge-density wave (CDW) riodic potential. If a small dc field is applied near the
1
asfoundinmetaldichalcogenides[4]andinα-U[5]. Ura- threshold field, the particle may be displaced from the
0
nium is unique in the periodic table in that it is the only bottom of the well. For fields less than E the par-
6 th
0 elementknowntoexhibitaCDWintheabsenceofdistor- ticle remains localized within the potential. For electric
/ tions due to a SDW. Although detailed band-structure fieldsgreaterthanE theparticleslidesdownthepoten-
t th
a calculations on the CDW transitions have been carried tial leading to an extra contribution to the conductivity
m
out[6],knowledgeoftheCDWenergeticsinα-Uremains generated by motion of the collective mode through the
- to be experimentally verified. crystal, or Fr¨ohlich conduction.
d
Across the light actinide elements (thorium – pluto- Significantdifferencesinsingleandpolycrystallineura-
n
o nium) itinerant5f-electronslower their energies by caus- nium have been observedin the thermalexpansion,high
c ing Peierls-like distortions [7], leading to the lowest- temperatureneutrondiffractionandcalorimetry[10,11].
:
v symmetry structures in the periodic table. Like their The temperature dependence of the thermal expansion
i quasi-one- and two-dimensionalcounterparts, the collec- coefficients for single crystal uranium differ in the three
X
tive mode in the crystal is characterized by an energy orthorhombicprincipledirections[5]. Presumablyasim-
r
a gap in the single-particle excitation spectrum originat- ple ensemble averagefor a polycrystalsample would dis-
ing from a charge modulation of the periodic lattice [8], playa temperature dependence different fromanyin the
ρ(x) = ρo + ∆ρ cos(2kFx+φ),whereρo isthe density single crystal. Therefore it was anticipated that a com-
ofthenormalstate,∆ρistheamplitudeoftheCDW,kF parison of the single- and polycrystalline specific heat
istheFermiwavevector,andφisaphasefactordefining would provide a useful probe of the effects on the Debye
the position of the periodic modulation. spectrumassociatedwithmicrostructuraleffects. Inthis
In a perfect lattice, a CDW should be able to slide Letter we show that the differences between the specific
throughout the crystal without resistance like a super- heat of crystals that undergo the weakly pinned CDW
conductor,assuggestedby Fr¨ohlich[9]. However,inreal transitions (single crystals) and those that are strongly
crystalspinningbydisorderfrommicrostructure(defects, pinned CDW (polycrystals) provide a definitive way to
twins, grain boundaries, impurities, surfaces, etc.) ele- determine the energetics of pinning.
2
Biljakovicet al.[12]havedescribedtheeffectofcollec-
tive mode dynamics on the phonon spectrum. In these
cases, a CDW modifies the phonon spectrum, which can
be described by a mixture of acoustic and optic modes.
The modified Debye spectrum features two cutoff fre-
quencies: a lower frequency corresponding to a pinned
state,hν =k T ,andanupperfrequency,hν =k Θ ,
o B o φ B φ
corresponding to the normal Debye temperature for the
collective modes [12]. With this modification, the excess
heatcapacity C (i.e. the difference betweenthe specific
φ
heatsofthesingle-andpolycrystal,C =C −C )
φ single poly
can be written as [13]
Θφ/T
Cfit =3N k T 3 dx x−x 2 x2 ex , (1)
φ φ B (cid:18)Θφ(cid:19) Z (cid:0) o(cid:1) ex−1 2
To/T (cid:0) (cid:1)
where we have introduced the notation x = hν/(k T),
B
with k being the Boltzmann constant, and N denotes
B φ
the number of excitations involved in the transition.
Single crystals of uranium were formed from electro-
transport in a molten LiCl-KCl eutectic electrolyte con-
tainingUCl atapproximately3wt%,asdescribedelse-
3
where[14]. Afterthe singlecrystalsweremeasured,they
FIG.1: (Coloronline)MeasuredspecificheatplottedasC/T
werecastthroughinductionmeltingofthesinglecrystals vs. T. The transitions at 38 and 41 K are broadened in the
inaBeOcrucible,underaninertatmosphere,toprepare polycrystal,asthespecificheatforthepolycrystalrisesmore
polycrystalline samples of the same pedigree. The sam- quicklybeforetheonsetofthetransitioninthesinglecrystal.
ple was melted only once to minimize the possibility of The inset shows the temperature dependence of the excess
entropy∆S =S −S .
contamination from the crucible or the carrier gas, then φ single poly
cleaned in concentratednitric acid. The grainsizes were
20-50 microns. The low-temperature specific heat of the
samples were measured using a semi-adiabatic calorime- metric measurements of Manley and coworkers to arise
ter [15] in the temperature range ∼0.5 to 100 K. The from a microstrain contribution at the grain boundary
single crystalhadamass of0.5g andthe polycrystalline interfaces in polycrystalline uranium [11].
sample mass was 1.4 g. Aswithfamiliar1Dand2DCDWs,weanticipatethat
The effect of microstructure on collective mode pin- the CDW transitions correspondto a lock-incommensu-
ning is shown in Fig. 1. The C/T versus T curves for rate with the periodicity of the underlying lattice, sepa-
polycrystal uranium and single crystal show a different rated by mixed phases of commensurate spatial regions
contribution throughout the temperature range encom- and discommensurations [16]. Phonon softening in α-
passing the CDW state. Specifically, three CDW transi- U has been observed below 50 K [17]. The new unit cell
tionsareobservedinthesinglecrystalat23K(α3),38K doublesinthe[100]direction(adirection)atTα1 =41K,
(α ), and 41 K (α ). While the polycrystalline sample becomes six times larger in the [010] direction (b direc-
2 1
showsno sharpexcessheatcapacityinFig. 1,onesees a tion)atTα2 =38K,andbecomesnearlysixtimeslarger
significantdifference inthe single-crystal sample. Start- in the [001] direction with the same periodicity (com-
ingbelowtheα3transition,theheatcapacityofthepoly- mensurate) along the b direction, at Tα3 = 23 K [5].
crystallinesampleislarger. Theheatcapacityislowerat Consequently, the volume of the new cell is 6000 ˚A3.
the α transitionreturningto a largervalue upon warm- Pinning of the collective modes by microstructure, also
3
ing to temperatures between the α and α transitions. consistentwiththermalexpansionmeasurements,results
3 2
Similarly it returns to a lower value through α and α in the polycrystal exhibiting a minimum in (∆L/L) be-
2 1
transitions. Thisexcessatthetransitionsisshowninthe tween 45 and 50 K. The relative volume change between
excessentropy,∆S =S −S ,depictedinthein- 4and50Kissignificantlylargerinsinglecrystals[5,10].
φ single poly
set of Fig. 1a. The excess contributions are consistent Furtherevidenceforthecollective-modepinningispro-
with the formationof an energygapin the single crystal videdbythelinearregionofthelow-temperaturespecific
thatismuchlargerthanthatobservedinthepolycrystal. heat (see Fig. 1b), where we notice a pronounced dif-
AbovetheCDWtransitionsthedifferenceinentropyhas ference in the low-temperature Debye limit. Figure 1b
been determined through neutron diffraction and calori- shows an expanded view of C/T vs. T2. Below 10 K,
3
each data set was fit to the electronic and lattice spe-
cific heat
C/T = γ + β3 T2. (2)
The Debye temperature, Θ , was obtained from β , as
D 3
ΘD = 3 12π4R5β3, (3)
q
(cid:14)
while the y-interceptgivesthe electronicspecific heat, γ.
The parametersobtainedfromthe fitto Eq.(2),andthe
inferred Debye temperatures are listed in Fig. 1b. The
most striking difference between the two samples is that
the slope, β , increases by a factor of 3 in the polycrys-
3 FIG. 2: (Color online) Excess specific heat plotted as C /T
φ
talline sample, resulting in a 35% reduction in Θ [18].
D vs. T. Note, the excess specific heat can be negative since
The large differences in the specific heat of the single- we neglected the pinning of the collective mode in the single
crystal and polycrystalline samples between 15 K and crystal. ThelinesdepictCfit/T inthepeakregion,whereCfit
φ φ
60 K (Fig. 1a) and the low-temperature limiting De- iscalculated from themodified Debyelaw, Eq.(1),with the
bye temperatures (Fig. 1b) suggest a modified Debye individualpinning frequencies shown aboveeach transition.
spectrum, as in Eq. (1). The excess heat capacity, C ,
φ
as a function of temperature is depicted in Fig. 2. In
the b direction, which is possible because there is elas-
other words C represents the heat capacity of the col-
φ
ticenergyavailabletofurtherlowertheelectronicenergy
lective modes above their respective pinning frequen-
(see inset in Fig. 1a).
cies. Anomalies such as these are known to arise from
The estimated values of the pinning frequency are
the phonon-branch splitting into an optic mode and an
lowerthanthoseobtainedforNbTe (540GHz)andKCP
acoustic mode below the CDW condensate formation. 4
(200GHz)butsimilarto(TaSe ) I(42GHz),wherenon-
SuchfeaturescannotbereproducedfromanormalDebye 4 2
linear conduction due to CDW motion is observed [21].
modelorDebye/Einsteinmodels,andinordertoanalyze
Fromthepinningfrequencies,weestimatealowerbound
the three anomalies, it is necessary to consider the two
on the threshold electric field [8], as
different Debye temperatures obtained in Eq. (3) and to
fit the excess specific heat, C , using the integral evalu-
ated at temperatures near theφtransitions. This descrip- Eth = 12 m∗ωo2ekF , (4)
tion of the modified Debye spectrum is further evident (cid:14)
from the fact that ΘD obtained from the single crystal where m∗ is the effective mass and νo = ωo/(2π) is the
is the only calorimetricvalue [15] to match the value ob- pinning frequency obtained from the fit of the excess
tained from elastic constant measurements [20]. specific heat capacity to Eq. (1). With m∗k = 16.6
F
Keeping Nφ and νo as free parameters, and Θφ fixed × 10−30 eV s2/atom, one obtains Eth as 14.5, 89, and
to the single-crystal value listed in Fig. 1b , the low- 133 V/cm for the α , α , and α transitions, respec-
1 2 3
temperature fit for the polycrystalline sample leads to tively. Here,we haveapproximatedthe period(π/k )of
F
the following Nφ/NA values: 0.25 for α1, 0.55 for α2, the CDW, by 2a in the case of the α1 transition, and 4b
and 0.9 for α3 (NA is the Avogadro number). The val- in the case of the α2 and α3 transitions.
ues obtained for the pinning frequencies are indicated in Inlightoftheabovediscussion,wesuggestthattheso-
Fig.2foreachtransition[22]. Thesevaluesareconsistent lution of the long-standingdiscrepancies in α-U between
with electronic structure calculations [6], but pinning to the Debye temperatures obtained from calorimetry and
microstructure at α occurs by the opening of two gaps
1
in the Fermi surface, accompanied by a doubling of the
unit cell in the a direction. In essence, the two areas of
the Fermi surface with the highest density of states are
nested in the a direction. Because there are two energy
scales involved in this transition (the lattice distortion
energy and the electronic energy) and because there is
a nesting in the b direction only 2 K lower than the α
1
transition, it is apparent that the α transition is neces-
2
sarytoovercometheoveralllatticedistortionenergy: the FIG. 3: (Color online) Estimated electric field threshold en-
doubling ofthe unit cellinthea directionat41K leaves ergiesforα1 andα2 transitionsinα-Ushownwithsomecom-
the Fermi surface in a favorable geometry for nesting in mon quasi-one and two dimensional CDW systems.
4
Insummary,wehavemeasuredthespecificheatdiffer-
ence between single-crystal and polycrystalline α-U and
have described the excess heat capacity as the contribu-
tion of the collective modes above their respective pin-
ning frequencies. Below the CDW melting temperature
of 41 K, the phonon splits into a mixture of acoustic-
and optic-like phonon modes as evidenced by the differ-
ence in Debye temperatures, 256 K for the single crys-
tal and 184 K for the polycrystal. The long-standing
discrepancy between Debye temperatures obtained by
calorimetry and elastic constants, can be attributed to
the degree of pinning of the collective mode. The com-
mensurate transition at 23 K, exhibits sluggish kinetics,
likely due to topological defects present at low tempera-
FIG. 4: (Color online) Dependence of the commensurate
tures. Similar collective phenomena are likely to exist in
CDWtransitionwithcoolingtime. Thetransitionisapparent
only if the sample has been cooled slowly (1K min−1). With otheractinides. Recently,aKohn-likeanomaly[28]inthe
fastercoolingtheα3transitionisnolongerresolved,blending TA1 [011] branch, with softening of the [111] transverse
in with thebackground. modes, has been observedin the vibrationalspectrum of
fcc-stabilized plutonium [29, 30], and the formation of
′
the α-martensite lattice instability has been detected in
elastic constant measurements depends on the degree of the low-temperature specific heat [31].
pinning of the collective mode. It is interesting to com- This work was carried out under the auspices of
pare these energies for α-U with other well known CDW the U.S. Department of Energy. The authors thank
materials, see Fig. 3. At the lowestenergy, NbSe3 repre- P. S. Riseborough, G. H. Lander and M. F. Hundley for
sents the most widely studied CDW material [23]. Here, useful discussions.
the single-particle energy gap was studied as a function
ofdefectconcentrationbysubjectingthesamplestoradi-
ationdamage. Thethresholdfieldincreaseslinearlywith
defectconcentrationsupportingstrongpinningsimilarto
TaSe [24] and K MnO [23]. In α-U the CDW state [1] R.E.Peierls, Quantum Theory of Solids (OxfordUniver-
3 0.3 3
melts atT =41K (α )givingafrequency of75GHz for sity Press, 1955).
1
[2] InitiallyproposedbyM.L.BoriackandA.W.Overhauser,
the collectivemode. Here the doublingofthe unitcellin
Phys. Rev.B 18, 6454 (1978).
the a direction cuts the Brillouin zone in half forming a
[3] R.Pynn,W.Press,S.M.Shapiro,andS.A.Werner,Phys.
gap with energy 14.5 V/cm. Similarly, the α2 transition Rev. B. 13, 295 (1969).
occurs by nesting of the Fermi surface with less density [4] G. Gru¨ner and A. Zettl, Phys. Lett. 119, 117 (1985).
of states in the b direction. At α the collective mode is [5] G.H. Lander, E.S. Fisher, and S.D. Bader, Adv. Phys.
3
confined to the bottom of the well. Hence, the pinning 43, 1 (1994).
frequency and the threshold electric field are higher. [6] L. Fast et al., Phys.Rev.Lett. 81, 2978 (1998).
[7] Below thetransition thermally activated carriers remain
The confinement of the collective modes to GHz fre-
so that no complete Peierls insulating state is formed.
quencies in α-U is a result of topological defects com- [8] G. Gru¨ner, Rev. Mod. Phys. 60, 1129 (1988).
bined with low transformation temperatures. This ef- [9] H. Fr¨ohlich, Proc. Soc. London., Ser A 223, 296 (1954).
fect is clear in the α3 transition where sluggish kinetics [10] J.C. Jousset, ActaMetall. 14, 193 (1966).
are observed, as shown in Fig. 4. Upon slow cooling [11] M.E. Manleyet al.,Phys. Rev.B. 66, 024117 (2002).
fromtheincommensuratestate(35K)to20K,andthen [12] K. Biljakovic et al., Phys.Rev.Lett. 57, 1907 (1986).
[13] InapplyingEq.(1)totransitionmetalchalcogenides[12]
measuring on warming, the transition appears as a sym-
theexcessspecificheat,C ,wasdefinedasthedifference
metric bump. In contrast, when the sample is cooled 10 φ
between the specific heat of the sample and the Debye
timesfaster,thetransitionisnotdetected. Hystereticef-
specific heat.
fects are known to be more pronounced with decreasing [14] C.C. McPheeters, E.C. Gay, P.J. Karell, and J. Acker-
temperatures. As pointed out by Gru¨ner, nonlinear I-V man, JOM. 49, N7 (1997).
measurements in NbSe , the high temperature onset is [15] J.C. Lashley et al.,Phys. Rev.B. 63, 224510 (2001).
3
accompanied by oscillatory phenomena and evolves into [16] P.BakandV.J.Emery,Phys.Rev.Lett.36,978(1976).
[17] H.G. Smith and G.H. Lander, Phys. Rev. B. 30, 5407
a hysteresis behavior at low temperature [25]. This be-
(1984).
havior has been attributed to extended pinning centers [18] Usingthebandstructureγ =5.89 mJK2 mol−1 [19],
b.s.
or topologicaldefects [27]. The effect of slow time scales the electron-phonon parameter, λ = γb.s./γ −1, is de-
on the collective dynamics of CDWs has been addressed termined to be 0.5 for the single crystal, and 0.7 for the
by Myers and Sethna [26]. polycrystalline sample.
5
[19] Band structure calculations for α-U were performed us- [24] G. Mih`aly and L. Mih`aly, Phys. Rev. Lett. 52, 149
inganAPWmethodwithaddedlocalorbitals(P.Blaha (1984).
et al., WIEN2K, An Augmented Plane Wave Plus Lo- [25] A.Zettle,M.B.Kaiser,andG.Gru¨ner,SolidStateCom-
cal Orbitals Program for Calculating Crystal Properties mun. 53, 649 (1985).
(TechnischeUniversit¨at,Wien,Austria,2001),fortheex- [26] C. R. Myers and J. P. Sethna, Phys. Rev. B., 47, 171
perimentallattice parameters a =2.858 ˚A,b = 5.876 ˚A, (1993).
c = 4.955 ˚A. [27] S.E.Brown,G.Mozurkewich,andG.Gru¨ner,SolidState
[20] E.S.Fisher and H.J. McSkimin, J. Appl.Phys. 29, 1473 Commun. 54, 23 (1985).
(1958). [28] W. Kohn, PhysRev.Lett. 2, 393 (1959).
[21] S.Mori et al.,Physica B 329-333, 1298 (2003). [29] J. Wonget al.,Science 301, 1078 (2003).
[22] A direct experimental measurement of the pinning fre- [30] R.J. McQueeney et al., Phys. Rev. Lett. 92, 146401
quencies, νo, and of the number of excitations, Nφ, are (2004).
necessary to validate thevalues quoted here. [31] J.C. Lashley et al.,PhysRev.Lett. 91, 205901 (2003).
[23] W.W. Fuller, G. Gru¨ner, P.M. Chaikin, and N.P. Ong,
Phys.Rev.B 23, 6259 (1981).
