Table Of ContentNMR and Electronic Structures in Type-I BaAlGe Clathrates
Weiping Gou,1 Sergio Y. Rodriguez,1 Yang Li,2 and Joseph H. Ross, Jr.1
1Department of Physics, Texas A&M University, College Station, TX 77843-4242
2Dept. Engineering Science and Materials, Univ. Puerto Rico at Mayaguez, Mayaguez, PR 00681-9044
(Dated: January 29, 2016)
We describe 27Al NMR experiments on Ba8AlxGe46−x type-I clathrates coupled with ab initio
computationalstudies. Forx=16,calculatedspectradeterminedbytheabinitioresultsgavegood
6 agreement with the measurements, with best-fitting configurations also corresponding to the com-
1 putedlowest-energy atomic arrangements. Analysis of the NMRresults showed that adistribution
0 of Knight shifts dominates the central portion of theline. Computational results demonstrate that
2 this stems from the large variation of carrier density on different sites. Al-deficient samples with
x = 12 and 13 exhibited a split central NMR peak, signaling two main local environments for Al
n
ions, which we connected tothe presence of vacancies. Modeling of thewide-line spectrum for x =
a
12indicates a configuration with more Alon the24k site than for x= 16. The resultsindicate the
J
8 importance of non-bonding hybrids adjacent to the vacancies in the electronic structure near EF.
Wealso address thestatic distortions from Pm3n symmetry in these structures.
2
]
i INTRODUCTION SAMPLE PREPARATION AND
c CHARACTERIZATION
s
-
l
r
t Samples of nominal composition Ba8AlxGe46−x were
m
prepared from the elements, mixed according to the de-
In recent years clathrates have gained consider-
. sired composition, by arc melting in argon following by
at able attention due to their thermal and thermoelectric annealingat500◦Cfortwodays. Samplesaredenotedby
m properties, and potential for device application [1–5].
Al ,forexampleAl representingBa Al Ge . Powder
x 16 8 16 30
Clathratesconsistofcagesofsilicon,germanium,ortinin
- x-raydiffractionmeasurementsshowedtheAl andAl
d a crystalline framework,with guest atoms located inside 13 16
samples to be pure type-I clathrate with no additional
n thecages. Thetype-Iclathrateshavethegeneralformula
o M X , where M and X represent the guest atoms and phases within experimental resolution, while Al12 and
c 8 46 Al alsoexhibitedasmallamountofdiamond-phaseGe.
10
[ framework atoms, respectively. Each unit cell contains
Rietveld analysis yielded best fits of Ba Al Ge
8.0 15.1 27.8
two 20-atom cages which form pentagonal dodecahedra,
1 for the nominal Al16 sample, and Ba7.0Al12.6Ge33.5 for
andsix24-atomcageswhichformtetrakaidecahedra. Re-
v Al . For Al these yielded 2.0, 5.2, and 7.9 Al atoms
12 16
0 cently much work has been devoted to Ba8Ga16Ge30, on the 6c, 16i, and 24k framework sites respectively in
4 which exhibits a particularly large thermoelectric effi-
the type-I structure (space group Pm3n, #223), and for
6 ciency at high temperatures [6]. An important issue has
Al , 1.5, 2.0, and 9.1 Al atoms. Lattice constants grad-
7 12
beentounderstandthedisplacementofBaionsinsidethe
0 ually decreased from 10.852 ˚A for Al16 to 10.838 ˚A for
larger cages,and the strong resulting effect on transport
. Al and 10.814 ˚A for Al . Resistivities exhibited posi-
1 behavior. 12 10
0 tivetemperaturecoefficients,typicalofmetalsorheavily-
6 doped semiconductors, with, for example, Al16 having a
1 low-temperature value of 170 µΩ-cm, within the range
:
v It was recently shown that Al alloying in observed previously for Ba8Al16Ge30 samples [8]. The
Xi cBoan8dGuac1t6iv−ixtAylx[7G],e3h0owceavner aitctwatsoaltsuoneshotwhne btyhehrimgha-l Ainld1e2pseunsdceenpttibvailliutye oisf d−ia1m.6a×gn1e0t−ic3, wemituh/ma otel;mspimerialaturrteo-
r
a resolution crystallography that in Ba Al Ge the other Ge clathrates [9] this is larger than the summed
interactions between Ba and group-III A8l io1n6s st3r0ongly corediamagneticcontributions(−0.5×10−3inthiscase).
affect the vibrational behavior of cage-center ions A feature of Ba8AlxGe46−y is the occurrence of spon-
[8]. An additional feature which strongly affects the taneous vacancies to balance the electron count in the
electronic behavior is the relative ease of breaking the framework, according to the Zintl concept [10]. For
Ge framework, with vacancies spontaneously formed to example, for Al the nominal Zintl composition is
12
maintain electronbalance in off-stoichiometrymaterials. Ba Al Ge (cid:3) ((cid:3) indicating a vacancy), since if each
8 12 33 1
Thus it is interesting to examine the electronic behavior Ba donates twoelectrons andAl has valence 3, there are
of these materials using a local probe spectroscopy. In four excess framework electrons per cell after filling all
this paper,we reporta study ofBa8AlxGe46−x withx = bonds, enough to fill the nonbonding orbitals adjoining
12–16, examining local variations in electronic behavior a single vacancy. The expected result is a semiconduc-
as the composition is varied and vacancies are formed. tor or low-carrier density material. Ba Al Ge has a
8 16 30
2
balanced framework electron count and no spontaneous
vacancies, while as the Al content decreases the vacancy
countwouldincrease,thelimitingcasebeingBa Ge (cid:3) ,
8 43 3
for which the vacancies form an ordered superstructure
[11–13]. Such behavior is also seen in Ba8Al16−xSi30+y
[14, 15], however vacancies form less readily in the Si
framework, for example Ba Si remaining a strongly
8 46
metallic superconductor [5, 16]. For Ba8GaxGe46−y the
vacancy count was determined to be about half the ex-
pected Zintl value over a range of compositions [17],
while scans of our nominal Al sample by wavelength
10
dispersion spectroscopy (WDS, an electron microprobe
method), yieldeda uniformclathrate phasewith compo-
sitionBa Al Ge ,thusindicatingthreevacanciesper
8 9.2 33.6
formula unit, similar to the number present in Ba Ge ,
and somewhat larger than the expected number8for43a FIG. 1: (Color online) NMR central lines for Ba8AlxGe46−x,
x = 12, 13, and 16, at room temperature.
Zintl phase. We saw no superstructure peaks in powder
XRDthatmightindicatelong-rangevacancyorderingas
seen in Ba Ge .
8 43
NMR experiments were performed at a field of 8.9
T using a pulse spectrometer described previously [18].
27Al NMR shifts were calibrated using dilute aqueous
AlCl as zero-shift reference. Samples were powdered
3
and mixed with KBr as an insulating medium. Magic
AngleSpinning (MAS) measurementswereperformedat
roomtemperatureusingaBrukerAvance400MHzNMR
instrument, with Al O as reference.
2 3
NMR MEASUREMENTS
NMR spectra of samples Al , Al , and Al mea-
12 13 16
sured by spin echo integration at room temperature are
shown in Fig. 1. For Al and Al we observe two ob-
12 13
vious peaks, while for Al a single, broader line is seen.
16 FIG. 2: Full NMR lineshape for Ba8Al12Ge34 from a spin-
These central lines are surrounded by a more extended echo measurement at 4.2 K. Inset shows central line portion
resonance signal, mapped out in detail at lower temper- of the spectrum.
atures, as shown in Fig. 2 for Al . The central lines
12
correspond to ∆m = 1/2 to –1/2 transitions for the I
= 5/2 27Al nucleus, while the broad background is due line like that of the static spectrum, flanked by spin-
to a superpositionof the other four transitions. The two ning sidebands in place of the near portions of the non-
peaks in the Al central line overlap more strongly in central resonances. Note that the paramagnetic shifts
12
Fig. 2 than in Fig. 1 due to relative changes in NMR increase to the left in this figure, the reverse of the ori-
shift vs. temperature. The valleys covering about 50 entation in Fig. 1. The further wings of the quadrupole
kHz on either side of the central line are artifacts due to powder pattern appear in the spin-echo measurements
differences in pulse response for the five transitions. For (Fig. 2) but fall out of range of the MAS measurement.
the type-I structure, the three framework sites all have Al yielded a similar result, though with sidebands less
13
someAloccupationasindicatedbythe x-rayfits, andin distinct indicating increasedinhomogeneous broadening.
addition there are distinct local neighbor configurations Thespinningfrequencydependenceofthesidebands(in-
due to the varied occupation of these sites. Thus the set,Fig.3),showedthecentertobe240ppm,underneath
NMRlinesaresuperpositionsofsignalsfrommultiplelo- the low frequency peak. Thus, the near portions of the
calenvironments,with their individual powder patterns. non-centrallinesareassociatedwiththispartofthe cen-
AspinningspectrumforAl isshowninFig.3. Magic tral line spectrum. For Al MAS revealed a broadened
12 16
AngleSpinning(MAS)cannarrowfirst-orderquadrupole central-linelikethatofthestaticspectrum,againflanked
powderpatternssuchasthebroadsatellitelinesseenhere by spinning sidebands. A spinning-frequency analysis
[19]. The Al MAS spectrum has a two-peaked central (not shown) showed the non-central background to be
12
3
FIG.3: SpinningNMRmeasurementofBa8Al12Ge34atroom
temperature,with14kHzrotorspeed,andhigherfrequencies
to the left. Inset: rotor speed dependence shows that broad
backgroundcorresponds tolower-shift peak. Notereversalof
paramagnetic shift axis orientation relative to other figures.
FIG. 4: (Color online) Frequency dependence of T1 for
Ba8Al12Ge34 compared to central line shape (lower curve).
Solid curve with monatonic decay is T1 calculated according
centered at 330 ppm, near the center of the featureless
to Korringa law, described in text.
central line for this sample, and somewhat higher than
for Al .
12
To further understand the central line shifts, we mea-
suredthefrequencydependence ofT ,theNMRlongitu-
1
dinalrelaxationtime,atroomtemperature. FortheAl
16
sample (Fig. 4), T decreases gradually with increasing
1
frequency, whereas for Al (Fig. 5) there is a plateau
12
coinciding with the lower-frequency peak in the central
line. At the upper end both systems follow a Korringa
law for T vs. shift, indicating the shift to be dominated
1
byaKnightshift(K). TheKorringalawmaybewritten
[19, 20],
~ γ2
K2T T =K−1(α)S ≡K−1(α) e, (1)
1 4πk γ2
B n
where γ and γ are the electron and nuclear gyromag-
e n
netic ratios, respectively, and K−1(α) represents a mul- FIG. 5: (Color online) Frequency dependence of T1 for
tiplierwhichamongotherthingsmaybeduetoelectron- Ba8Al16Ge30 compared to central line shape (lower curve).
electron interactions. In simple metals it is commonly Solid curve with monatonic decay is T1 calculated according
observedthat K−1(α) has a value in the range1–5. The to Korringa law.
curvesinFigs.4and5weredrawnbasedonEqn.1,with
K−1(α)=2.7and5.5,respectively. Thisbehaviorpoints
to aninhomogeneousdistribution of Knightshifts as the ity terms [20]) may also contribute to the total shift.
mainfactordeterminingthehigher-frequencyportionsof Forcomparison,thegroupIII-Vzinc-blendesemiconduc-
both lines. For Al the lower peak exhibits a plateau in torsAlP, AlAs, andAlSb exhibit paramagneticchemical
12
T indicating some other source of broadening for that shifts from 70 to 140 ppm [21, 22]. Shifts of this order
1
portion of the line. Measurements on Al yielded simi- of magnitude could be responsible for a portion of the
13
lar results to those of Al . The presence of an inhomo- Korringa enhancement identified above.
12
geneous distribution of isotropic shifts explains why the Notethatfor27AlNMRinBa8Al14Si31 [15],threesep-
MAS spectra are little narrowed compared to the static arated peaks could be assigned to the three framework
spectra,sincesuchadistributionisnotnarrowedthrough atom types, with 6c having a rather distinctive shift.
spinning. That is not the case here, instead the individual sites
Central line shifts also include a second-order overlapgivingalinethatisasuperpositionofframework
quadrupole contribution; as shown by the simulations sites. A calculation described below shows for a model
below, this term generally represents a few percent of configuration how this results from a large distribution
the total shift for these central lines, except for the up- of Knight shifts producing the central line shape.
per peak in Al where it is somewhat larger. A chemi- (T T)−1 values determined at three temperatures, for
12 1
cal shift (due to T-independent Van Vleck susceptibil- peak positions of the Al and Al central lines, are
12 16
4
tureshavingdifferentatomicarrangements,startingwith
atomic positions of the reported parent Ba Ge Al
8 30 16
structure [28]. Where justified by initial results we also
usedWIEN2ktominimizeinternalstructuralparameters
before finalizing the results.
We obtained EFG’s for each Al site from the calcu-
lated all-electron charge distribution, using a method
usedinrecentyearswithconsiderablesuccessforanalysis
of NMR spectra [29, 30]. Relations for NMR transition
frequencies between the mth and (m + 1)th levels in a
single crystal are well established [20], with quadrupole
shifts given in terms of the EFG principal values (∂2V,
∂x2
∂2V, ∂2V), η = (∂2V − ∂2V)/∂2V, and ν = 3e2qQ ,
∂y2 ∂z2 ∂x2 ∂y2 ∂z2 Q 2I(2I−1)h
with eq = ∂2V the largest principal value and Q the
∂z2
FIG. 6: (Color online) T1T reciprocal product for Al12 and nuclear quadrupole moment. From computed EFG ma-
Al16 samples at peaks of the central lineshapes. Horizontal trixelements the quadrupolelineshape wassimulatedby
lineindicatesKorringabehavior. Inset: K2T1T productmea- summing a contribution from each m to m +1 transi-
suredatthesamepeakpositions,withsamesymbolsasmain tion, with a weighting factor p(I +m)(I −m+1) for
plot. Horizontalline: Korringarelationforbare27Alnucleus. each m, and using standard expressions for the angu-
Dashed lines are guides to theeye.
lar dependence [20]. The results were used to calculate
the powder patterns numerically, by summing uniformly
overall solid angles,including the first and second-order
shown in Fig. 6. (T T)−1 is constant at higher tempera- quadrupoleshiftforeachtransition. Forcentrallines,the
1
turesforthemainpartsofbothlines,indicatingKorringa only significant quadrupole contribution is the second-
behavior (horizontal line in the figure), but deviates up- order term, whereas the shift of the broad background
wardat 4 K. The upper Al peak partially merges with is dominated by first-order quadrupole shifts. In some
12
the lower peak and thus could not be assigned a shift cases an anisotropic chemical shift was also included in
at low temperatures. In the inset of Fig. 6 is shown the the angular summation, although this did not provide
K2T T product, using for K the total measured shift. apparentimprovementandhence was not used in any of
1
With K−1(α) = 1 indicated by the solid line, K2T T is the final simulations.
1
enhancedforbothmaterialsasdescribedabove,andalso The 10 configurations considered for Ba Ge Al are
8 30 16
variesslowlywithtemperature(dashedlines),mostlydue summarizedin Table I, where the first three columns in-
to slowvariationsin K. Suchvariationsin K andT1 are dicatethenumberofAlatomspercellineachframework
not unlike those reported for the host resonance in Si:P site of the parent Pm3n structure, and the fourth shows
heavily doped above the metal-insulator transition, and thenumberofadjacentAl-Alpairspercell. Forthesesu-
may indicate that local composition fluctuations lead to perstructures we retained the cubic cell symmetry, leav-
an impurity band partially merged with the conduction ing 54 independent atoms (P1 space group), except for
band[23,24]. Similarbehaviorwasfoundfor71GaNMR a few cases allowing higher symmetry. 8 of the config-
in Ba8Ga16Ge30, while in Sr8Ga16Ge30 simple Korringa urations are the same as considered in reference [31] for
behaviorwasobservedoverawidetemperaturerange,as Ba Ga Ge , while in reference [32] a 3-4-9 configura-
8 16 30
in a simple metal [25]. tion was considered, similar to the structure determined
for Ba Si Al [15].
8 31 14
Computed NMR lineshapes could be distinguished by
COMPUTATIONAL METHODS AND theshapesofthenon-centralportionsofthelines,andwe
DISCUSSION
found that Al atoms situated in Al-Al pairs contributed
the largestEFG’s. Forconfigurationswith3ormoreAl-
Ab-initio calculations were carried out using the full- Al pairs this led to broadsimulated lines providing poor
potentiallinearizedaugmentedplane-wavemethodasim- agreement with the data. Thus we rule out configura-
plemented in the WIEN2k package [26], both for anal- tions with larger numbers of Al-Al pairs, a feature that
ysis of the electronic structure and for calculation of is verydifficult to determine via diffraction experiments.
the electric field gradients (EFGs) [27], with subsequent These configurations also had the highest energies; col-
numerical simulation of NMR lineshapes. In all cases umn 5 of Table I shows computed total energies per for-
we used the Generalized Gradient Approximation, with mula unit after parameter optimization, relative to the
thePerdew-Burke-Ernzerhofformalismfortheexchange- most stable configurations. Note that in this case only
correlation term. We considered 33 distinct superstruc- the internal parameters were optimized, leaving lattice
5
TABLE I: Al framework occupation used in calculations of
Ba8Al16Ge30,performulaunit,alongwiththenumberofAl-
Alpairspercubiccell,χ2 from thelineshapesimulation, and
therelative calculated energies per formula unit.
6c 16i 24k Al-Al χ2 Energy (eV)
3 1 12 0 2.9 0.00
3 3 10 0 2.3 0.60
3 4 9 1 8.0 0.00
3 4 9 2 1.6 1.03
4 2 10 2 5.5 1.28
5 3 8 4 6.4 2.31
4 6 6 3 3.6 –
5 1 10 7 5.4 –
6 4 6 5 4.6 –
6 0 10 10 3.9 –
constants identical. The last four configurations were FIG.7: SimulatedNMRspectraforseveralBa8Al16Ge30con-
figurations,togetherwithspin-echoNMRspectrummeasured
not optimized due to the long required computing time,
at4K.Inset: expandedviewofthecentral-transition region.
however the default parameters gave poor fits and rela-
tive energies greater than 2 eV, and for other configura-
tions optimization produced small changes in computed
larpatternofalternatingGeandgroupIIIatomsaround
lineshapes and in total energy (typically 0.2 eV).
each six-sided ring of the larger cages. Given its equiv-
Fig. 7 shows severalsimulations compared to the Al16 alent energy in our results, we would expect the 3-4-9-1
spin-echo spectrum. As in Fig. 2, the two valleys im- configurationto predominate at the growthtemperature
mediately adjacent to the central line are experimental due to its greater configurational entropy. Yet, this con-
artifacts. Amongstructuresconsidered,3-4-9-2provided figuration yielded a distribution of EFG’s too small to
the best agreement, while the 3-3-10-0and 3-1-12-0 also agree with the NMR lineshapes. Instead, 3-4-9-2, with
produced acceptable fits, the four numbers referring to one additional Al-Al neighbor pair, was among the best
the Al site occupations and number of Al-Al pairs, as fitting. (3-4-9-0 can also be formed, and it seems likely
shownleft-to-rightinTableI.Inthesimulationsaniden- that its calculated energy would be lower than either 3-
ticaldistributionofisotropicshiftswasaddedtoallsites, 1-12-0 or 3-4-9-1, but presumably with a poor fit to the
correspondingtothebreadthofthecentralline,andrep- NMR.) The 3-4-9-x configurations are in line with the
resenting a Knightshift distribution as evidenced by the general framework occupation rules established in ref-
T1 measurements. Additional center-of-mass shifts for erence [8] from neutron scattering, and also consistent
each site due to the second-order quadrupole effect are with our powder XRD refinement. Thus according to
relatively small; such shifts are responsible for the just- our results 3-4-9-2 is the best representationof the alloy
discernible frequency differences between curves in the configuration; it has more Al-Al pairs than energetically
insetofFig.2. The onlyotherparameterusedwasasin- favored,althoughthedifferenceissmallanditispossible
gle overallamplitude adjustment, the same for eachcon- that entropic processes or incomplete equilibration may
figuration,tomakethecalculatedcentrallinesmatchthe enhance the likelihood of this configuration.
experiment. This produced a good match for the entire Bandstructuresfor Al configurations3-1-12-0and3-
16
lineshape,withthe3-4-9-2simulationfallingparticularly 4-9-2 are shown in Fig. 8 (a) and (b). Both are semi-
closeto the data. The sumofsquaredifferencesbetween conducting, the latter in general agreement with a pre-
the data and each fit provide an indication of the good- viously report [32]. Our results for 3-4-9-1 and 3-4-9-2
nessoffit,asshownbytheχ2 (TableI).NotethatNMR, are almost identical. The three-fold minimum at M is
as a local probe, is much less sensitive to widely spaced nearly isotropic, with an effective mass 1.93 m for 3-1-
e
defects and surface effects that, for example, transport 12-0 (2.35 m for 3-4-9-2). Decomposing the calculated
e
measurements. This can explain why the fitting based densityatthisminimuminsidethemuffin-tinspheres,we
on computed results for perfect single crystals can work find average Al s-partial contributions of 0.16%/0.18%
well for polycrystalline samples. for 3-1-12-0/3-4-9-2 (percentages relative to the entire
Although3-4-9-2providedthe bestfit, 3-1-12-0and3- 54-atom cell). In the effective mass approximation, the
4-9-1 had nearly equivalent total energies that were the Pauli susceptibility (χ ) can be calculated for a given
p
lowest calculated for Al configurations. Of these, 3-1- electron density, since it is proportional to the density
16
12-0 was also the minimum energy configuration identi- of states[33]. Assuming conduction electrons introduced
fied in reference [31] for Ba Ge Ga , featuring a regu- extrinsicallythroughthepresenceofelectronicallyactive
8 30 16
6
defects, we cantherebyobtainthe correspondingKnight
shift using
χ V f H
p cell Als HF
K = , (2)
µ
B
where f is the calculated partial s fraction on a given
Als
Al site, andH is the s-hyperfinefield, takento be 1.9
HF
MG [20]. Taking the average central-line shift at room
temperature (0.0375%) to be a Knight shift, the com-
puted results yield n = 1.1 × 1021 cm−3 for 3-1-12-0,
corresponding to a small degenerate carrier pocket. 4.1
× 1020 cm−3 is calculated for 3-4-9-2. The calculated
carrier densities are in the same range as those found
foras-grownBa Ga Ge andSr Ga Ge [3],typically
8 16 30 8 16 30
attributedtosmalldeparturesoftheframeworkcomposi-
tion from the ideal semiconducting materials. Note that
Eqn. 2 will be valid only if enough carriers are present
to reachthe degenerateregime,whichcanbe verifiedfor
the densities found here. This result is also consistent
with the observed positive temperature coefficient of re-
sistivity. The wide-line NMR fits, which are dominated
by the EFG’s, will be affected very little by the small
density of such carriers. While these shifts are metallic,
it is not unreasonable that the Zintl concept may still
apply, since the carrier densities are thus the departure
from the semiconducting configuration are small.
Inadditiontotheaverageshift,therangeofcalculated
partial densities on individual sites is large and can also
FIG. 8: (Color online) Calculated electronic structures for
account for the measured NMR line broadening. This Ba-Al-Ge clathrate configurations. (a) and (b): Bandstruc-
is shown in Fig. 9, where by superposing lines for each tures for Ba8Al16Ge30, 3-1-12-0 and 3-4-9-2 respectively, no-
Al site shifted according to its calculated f (with a tation as defined in text; (c), (d), (e): Bandstructures for
Als
single vertical scaling parameter) we obtain an inhomo- Ba8Al12Ge34−y(cid:3)y, (cid:3)0 4-8-0-0-0, (cid:3)1 4-8-0-0-0, and (cid:3)1 0-0-
12-0-2 configurations respectively. (f): Density of states for
geneously broadened line with a slightly larger intrinsic
thelatter, includingpartial densities for Aladjacent to a va-
widththanthemeasuredcentralline. Resultsfor3-4-9-2
cancy,Ge (6c), Ba (2d), and thetotal scaled by 1/10.
are similar with a largerwidth approximatelytwice that
of the data. The detailed shape, with two main peaks,
is not representative of the data, indicating that the cal-
culated superstructures are only approximations to the
alloy configuration, however clearly the width of the ob-
servedNMRlineisanindicationofthelargevariationof
conducting states among local atomic configurations in
the alloy.
InmodelingtheAl alloyweconsideredthe16super-
12
structures summarized in Table II. The expected Al
12
ZintlcompositionisBa Ge Al (cid:3) ,howeverforanom-
8 31 12 1
inal Al composition we found more vacancies than ex-
10
pectedinWDSscans,soweconsideredarangeofvacancy
concentrations. In all cases, vacancies were placed only
on 6c sites, shown to be the preferred site for vacancies
in the type-I structure [11–13].
Several calculated spectra are shown in Fig. 10, along
withtheAl spin-echospectrum. The(cid:3) 0-0-12-0-2and FIG.9: (Coloronline)CalculatedNMRlineshapefromcom-
12 1
(cid:3)3 0-0-12-0-6 configurations (notation corresponding to pduentesdityK1n.1ig×ht10sh21iftcsmf−o3r,3co-1m-1p2a-r0edcotonfiAglu16rartoioonm,-twemithpecraartruierer
first 6 columns of Table II) provided a good fit, while
data(filledcircles). Calculated 6cand16iportionsareasin-
other configurations agreed poorly. The large difference dicated; remainder dueto 24k sites.
betweencalculatedspectra was due mostsignificantly to
7
TABLE II: Configurations considered in calculations of
Ba8Al12Ge34−y. Number of framework vacancies, Al frame-
work occupations, number of Al-Al pairs and number of Al
adjacent to vacancies are given per formula unit. Calculated
lattice constant and relative formation energy, per formula
unit of Ba8Al12Ge34, given for minimized configurations.
vacancy 6c 16i 24k Al-Al Al-vac a (˚A) energy (eV)
(cid:3)0 4 8 0 0 0 11.01 0.00
(cid:3)1 4 8 0 0 0 11.01 0.22
(cid:3)1 4 4 4 0 2 10.99 1.85
(cid:3)1 4 3 5 1 5
(cid:3)1 4 2 6 1 5
(cid:3)1 0 0 12 0 0 10.97 1.08
(cid:3)1 0 0 12 0 1 10.98 1.08
(cid:3)1 0 0 12 0 2 10.97 1.30
(cid:3)1 0 0 12 1 4 10.98 1.98
(cid:3)2 4 8 0 0 0 10.84 1.86
(cid:3)2 4 4 4 0 2
(cid:3)2 4 0 8 0 8 FIG. 10: Simulated NMR lines for several configurations of
(cid:3)2 4 3 5 1 5 Ba8Al12Ge34−y together with spin-echo NMR lineshape. In-
(cid:3)2 4 2 6 1 6 set: expandedview of the central transition region, with cal-
(cid:3)2 0 0 12 2 4 10.97 3.56 culated spectrum only for (cid:3)1 0-0-12-0-2 configuration with
(cid:3)3 0 0 12 0 6 10.95 5.03 one vacancy shown.
the large EFG for sites adjacent to a vacancy, ranging is reduced.
from ν = 1000 to 2800 kHz. Normalization was similar Comparedtothe best-fitting(cid:3) 0-0-12-0-2,othercon-
q 1
to that of the Al curves, with a single overall scaling figurationshavelowerenergy: (cid:3) 4-8-0-0-0is1.1eV/cell
16 1
used to force agreement in the central-line region. How- lower in energy and is the most stable calculated one-
ever,because ofthe two-peakedcentralline, twodistinct vacancyconfiguration. FillingthevacancywithGeyields
isotropic shifts were applied, with a portion of the sites (cid:3) 4-8-0-0-0with astilllowerenergy,althoughthe addi-
0
having a larger shift than the others, plus an additional tionalvacancy-formationentropyofthe(cid:3) configuration
1
overallbroadening function applied to the otherwise rel- maynegatethesmalldifference. Yetnoneofthe4-8-0-0-0
ativelynarrowcomputedcentral-line. Theshiftdistribu- simulationsagreewiththeNMRdata;the(cid:3) simulation
1
tionhasalmostnoeffectonthe broadnon-centralregion isshowninFig.10,andthatof(cid:3) issimilar,bothhaving
0
of the lines, and thus does not affect the fitting except EFG’s too small.
for the appearance of the central line. We will return to Band structures for some of these configurations are
consider a more detailed calculation of the Knight shift shown in Fig. 8. From the calculated densities of states
distribution below. for the (cid:3) 0-0-12-0-2 configuration (Fig. 8(f)) we can
1
Calculated total energies and lattice constants are calculate the site-dependent Knight shift, using Eqn. 2,
shown in Table II for minimized configurations. In the with the local susceptibility taken from the local Fermi-
formation energies we added the all-electron total en- leveldensityofstates,usingasimilarmethodasusedfor
ergy per atom for diamond-structure Ge, after volume theAl calculation. TheresultisshowninFig.11asthe
16
minimization using the same GGA-PBE formalism, to solidcurve,obtainedbysuperposingalineforeachofthe
compensate for each vacancy. Configurations not mini- 12 calculated Knight shifts, with an identical Gaussian
mized had simulated spectra differing significantly from broadening given to each line. Note that in contrast to
the data andrelative energiesgreaterthan2.5 eV before theAl case,forwhichasmallcarrierdensitywasadded
16
minimization. There are a number of configurations rel- to the otherwise insulating configuration, for this case
atively close in energy, however a general trend is seen there are no adjustable parameters, save for a vertical
with a preference for zero or one vacancy. Of the two scaling. The calculated peak at the right edge is due to
configurations best fitting the NMR spectra, (cid:3) 0-0-12- thetwoAlsitesadjoiningavacancyinthisconfiguration,
3
0-6hasaconsiderablyhighercalculatedenergy,while(cid:3) for which the local charge density is larger. Overall, the
1
0-0-12-0-2 is energetically more favored and has the ex- calculated shifts are larger than observed, although it is
pected single-vacancyZintl composition, thus provides a possiblethatthepseudogapatE mightbecomedeeper,
F
likely explanation for the data. This configuration also and hence the shifts smaller, if not for the well-known
agreeswiththetrendfromourXRDfitsindicatingAloc- tendencyforband-gapunderestimationinthePBE-GGA
cupationtransferredto 24k sites as the Al concentration method. Thus the agreement seems quite reasonable.
8
placements are consistently smaller that the experimen-
tal values [8]. The calculated results do not include dy-
namical terms, and also may reflect a balance between
opposing tendencies for different growth conditions, for
which displacements tended toward perpendicular crys-
tal directions [8], presumably due to different dominant
defect types. For Al configurationswith vacancies, the
12
calculated structural distortions are considerably larger;
all framework atoms next to the vacancy shift toward
the open space (by 0.35 to 0.55 ˚A), while the four Ba
6d sites in cages including the vacancy shift toward the
24k framework sites (Ge or Al) immediately adjacent to
the vacancy. This decreases the Ba 6d-(Ge/Al)24k bond
length from 3.64 ˚A to 3.39–3.43 ˚A. This is not quite as
shortasthesumofthecovalentradii(3.35–3.36˚A),how-
ever it is clear what develops is a much more localized
covalent bond involving the atoms adjacent to the va-
FIG. 11: (Color online) Calculated NMR line shape (solid cancy.
curve) from computed Knight shifts for (cid:3)1 0-0-12-0-2 con-
figuration, compared to Al12 room-temperature data (filled
circles).
CONCLUSIONS
Whilethecalculated(cid:3) 0-0-12-0-2configurationisnot
1
semiconducting, EF falls between bands over much of k Wide-line 27Al NMR measurements on Ba8AlxGe46−y
space (Fig. 8). In this way, this configuration might be clathratesyielded quadrupole-broadenedlineshapes that
consideredto satisfy the Zintl criterion,in which a semi- help to establish the local configuration of Al atoms in
conductor would be expected, with a hybridization gap these alloys. Combined with ab initio calculations of a
at EF. From the partial density of states [Fig. 8(f)] it number of superstructure configurations, we obtained a
can be seen that contributions from the two Al vacancy good fit to the NMR measurements for x = 16 that cor-
neighbors dominate the highest filled band, making up respondedtothemoststableconfigurationsidentifiedby
the peak just below EF, far outweighing contributions our computations. Analysis of the NMR and computa-
from any other site; the sites with the largest Ge and tional results showed that a large distribution of metal-
Ba contributions in this region give the two additional lic shifts dominates the central region of the NMR spec-
curves also plotted in Fig. 8(f). It is thus apparent that trum, indicating a considerable variation of carrier den-
theassociatedstatesarelargelynon-bondinghybridsad- sity within the cell. For x < 16 we found a splitting
joining the vacancy. Theconcentrationofsuchstatesfor in the main NMR peak, coinciding with the appearance
this band explains the larger Knight shift for such Al of vacancies in order to approximate a Zintl composi-
sites, and hence the observed split-off central peak. tion. Detailedanalysisforx=12yieldedaconfiguration
Having minimized the internal parameters for a num- giving very good agreement with the NMR spectrum.
berofconfigurations,wecanalsoaddresstheatomicdis- The fitting indicates that Al atoms tend to shift to the
placements which are of potential significance for ther- 24k framework site for x < 16. From an analysis of the
moelectric applications. In Al , calculated Ba displace- NMR shifts we deduced that Al non-bonding states play
16
ments from symmetry positions are very small even for a significant role near the Fermi edge. Calculations also
6d atoms identified through diffraction experiments as indicate rather small static distortions from the type-I
having distorted nuclear density profiles [8], and which structure for intrinsic Ba8Al16Ge30, but more significant
exhibit a low-energy ”rattling” mode [32]. In the 3-4- distortions with vacancies present.
9-2 configuration, the largest calculated Ba 6d displace-
mentis0.019˚A.Ofthe sixoptimizedAl structures,no
16
Ba 6d was more than 0.07 ˚A from the cage center, and
the overall average displacement was 0.025 ˚A. Also the
ACKNOWLEDGMENTS
longest framework bonds appeared in structures having
Al-Al neighbors, for example in the 3-4-9-2 configura-
tiontheAl-Albondsincreasedto2.61–2.77˚A,compared This work was supported by the Robert A. Welch
to the Ge-Ge bond lengths of 2.51–2.53 ˚A; this reflects Foundation, Grant No. A-1526, by the National Science
the weakness of the Al-Al bond. Calculated Ba 6d dis- Foundation (DMR-0103455).
9
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