Low temperature features in the heat capacity of unary metals and intermetallics for the example of bulk aluminum and Al$_3$Sc

3.1 MB·

by  Ankit Gupta

Low temperature features in the heat capacity of unary metals and intermetallics for
the example of bulk aluminum and Al Sc
3
Ankit Gupta,1 Bengu¨ Tas Kavakbasi,2 Biswanath Dutta,1 Blazej Grabowski,1 Martin
Peterlechner,2 Tilmann Hickel,1 Sergiy V. Divinski,2 Gerhard Wilde,2 and J¨org Neugebauer1
1Max-Planck-Institut fu¨r Eisenforschung GmbH, D-40237 Du¨sseldorf, Germany
2Institute of Materials Physics, University of Mu¨nster, D-48149 Mu¨nster, Germany
(Dated: January 25, 2017)
We explore the competition and coupling of vibrational and electronic contributions to the heat
capacity of Al and Al Sc at temperatures below 50 K combining experimental calorimetry with
3
highly converged finite temperature density functional theory calculations. We find that semilocal
7
1 exchange correlation functionals accurately describe the rich feature set observed for these temper-
0 atures, including electron-phonon coupling. Using different representations of the heat capacity,
2 we are therefore able to identify and explain deviations from the Debye behaviour in the low-
temperaturelimitandinthetemperatureregime30–50Kaswellasthereductionofthesefeatures
n
due to the addition of Sc.
a
J
PACSnumbers: 31.15.A,71.15.Mb

4
2
I. INTRODUCTION electronicexcitationsclosetotheFermisurface18–20. Re-

] cent finite-temperature density functional theory (DFT)
i
c calculations have indeed shown the importance of elec-
s The heat capacity is one of the key physical quantities
troniccontributionstotheheatcapacityforvariousmet-
- in materials research, since it can be directly measured
l als10,11,21–32. In all these works, however, the focus
r and has a significant technological and scientific impact.
t was on the high temperature regime (T > Θ ), where
m Itiscrucialtodeterminetheonsetofphasetransitions1,2 D
the importance of the electronic contributions also intu-
at. athnedCtoAcLoPnHstAruDcttetchhenrmiqoude3y)n,atmoicevdaalutaatbeamseasg(nee.tgo.,caulsoirnigc itively seems to be highest. While temperature effects
well below Θ are small on an absolute scale and have
m effects4,5, as well as for applications in microelectronics6 D
thus little effect on materials properties, analyzing the
and modern battery materials7. Computing the heat ca-
- asymptotic behavior of the heat capacity towards abso-
d pacityandunderstandingitsgenericfeaturesrequiresan
lute zero promises detailed insights into the various ex-
n accurate determination of the temperature dependence
o citation mechanisms. The careful analysis of the low-
of all relevant excitation mechanisms such as atomic vi-
c temperature regime, both with respect to the method-
brations,electronicexcitationsandthecouplingbetween
[ ologicaltoolsaswelltothephysicalphenomena,isthere-
them.
fore in the focus of the present study.
1
Recent advances in finite-temperature first-principles
v Expressingcalorimetricdatasolelyasasumofavibra-
9 calculations have led to an accurate description of key tionalphononcontribution(cubicinT)andanelectronic
9 thermodynamic quantities of a broad range of material contribution(linearinT)18,19 neglectscouplingsbetween
9 systems8–16. With the ab initio computed Gibbs energy
theseexcitationmechanisms. Theinterplaybetweenelec-
6 surface at hand, all measurable thermodynamic proper-
tronicandvibrationaldegreesoffreedomyieldsadiabatic
0
ties such as e.g. thermal expansion or temperature de-
as well as non-adiabatic corrections. The impact of elec-
.
1 pendent elastic constants for a given compound can be
tronictemperatureontheinteratomicforcesandthuson
0 accessed. Out of this set, the heat capacity C is par-
P phonon energies is typically added to the adabiatic part.
7 ticularly sensitive to fine details in the excitation mech-
Theterm“electron-phononcoupling”is(inthecontextof
1
anisms, since it is a second derivative of the Gibbs free
: perturbationtheory)reservedfortransitionprobabilities
v energy. Special care is required at low temperatures – betweenelectronicstatesduetoionicdisplacements33. It
i as demonstrated in the present work – where only the
X is described within the Eliashberg theory by a coupling
consideration of C /Tn with n = 0,...,3 allows a full
P strength,whichrescalestheelectronicheatcapacitybya
r
a evaluation of thermodynamic contributions to the heat mass enhancement parameter33–37. It is directly related
capacity in non-magnetic metals17 and alloys.
to the phonon linewidth arising from interactions with
The Debye model is commonly used to describe the electrons34. While electron-phonon coupling is an estab-
temperature dependence of the heat capacity of mate- lishedconceptinthefieldofsuperconductivity,itisrarely
rials at T Θ , (Θ : Debye temperature). Ac- discussed in calorimetry. Only few studies used this con-
D D
(cid:28)
cording to this model the heat capacity should exhibit tribution to explain observed deviations in heat capacity
a T3 dependence. Experimentally, however, in many measurements of simple metals when using only the two
metals deviations from the T3 behaviour are observed adiabatic contributions. In contrast to the present study
attemperaturesbelow50K.Thesedeviationscanbeat- the focus in these works was on temperatures well above
tributed to the realistic phonon dispersion as well as to the Debye temperature38,39. Though it is mathemati-
2
callyclearthatthelinear(i.e. electronic)termwilldom- A. Electronic contribution
inateoverthecubic(i.e. vibrational)termatsufficiently
lowtemperatures,adetailedanalysishowwellDFTcap- The electronic contribution Fel(V,T), which is partic-
turesthiscompetitionandthecouplingofelectronicand ularly important for the present low temperature study,
vibrational degrees of freedom at very low temperatures is obtained following Mermin’s finite-temperature DFT
is currently missing. Such a benchmark is in particular formalism22,63. Its physical origin is the thermodynamic
important for alloys, for which low temperature experi- excitation of electrons close to the Fermi energy (within
mental data are typically not readily available, making a range of approx. k T), which yields in lowest order of
B
finite-temperatureDFTcalculationsapromisingtoolfor the Sommerfeld expansion for the ideal Thomas Fermi
an accurate assessment of physical properties. gas a temperature dependence18,19
Inthepresentwork,weinvestigatetheaforementioned
deviationsfromtheDebyebehaviourinthelowtempera- CPel(T)=γT (3)
ture regime within a full ab initio approach and perform
with the Sommerfeld coefficient γ.
experiments as a benchmark. We take pure Al and the
The V,T parametrizationofFel(V,T)isdoneasfol-
intermetallic compound Al Sc as prototypical systems.
3 { }
lows: Firstly, we calculate the T-dependent part of Fel
The Al Sc precipitate phase is of prime importance to
3
by performing DFT calculations on an equidistant mesh
achieve the high mechanical strength of Al rich Al-Sc al-
of 11 volumes and 11 temperatures as
loysandhas,overthelastcoupleofdecades,enabledthe
impressive development of these alloys8,12,40–62. Besides
Fel(V,T)=Fel (V,T) E (V), (4)
possessingastructural(fcccrystalstructure)anddimen- tot − 0K
sional (lattice mismatch approx. 1.6% at 0 K, decreasing
where Fel is the total electronic free energy (including
with temperature) correspondence with the Al host ma- tot
the 0 K binding energy E ) obtained from a fully self-
trix,theseprecipitatesarecoherentlyincorporatedinthe 0K
consistent DFT calculation at a finite electronic tem-
Al host matrix and homogeneously distributed resulting
perature T corresponding to a certain Fermi smearing.
inafine-grainedmicrostructure,aswellashighyieldand
To obtain a dense temperature sampling for calculating
tensile strength. Since both, Al and Al Sc, share a com-
3 the heat capacity, we use a physically motivated fit22,
mon underlying face-centered cubic (fcc) lattice, with Sc
Fel(T)= 1TSel(T), where
substituting the corners of the conventional cubic unit −2
cell, we can directly connect the observed changes in the (cid:90)
heat capacity features with the chemical impact of Sc. Sel(T)= 2kB d(cid:15)Nel(T)[flnf+(1 f)ln(1 f)], (5)
− − −
with the Fermi-Dirac distribution function f = f((cid:15),T)
and with Nel(T) representing an energy independent
II. METHODOLOGY electronicdensityofstates. Thelatterisusedasafitting
quantity by expanding it up to a third-order polynomial
in T as (cid:80)3 c Ti with fitting coefficients c . Next, for
Toderivethethermodynamicpropertiesofagivenma- i=0 i i
the parametrization of the volume dependence of Fel, a
terialsystemwithinDFT,theHelmholtzfreeenergyisa
second-order polynomial is used to fit Fel(V). The vol-
common starting point. For a defect free non-magnetic
ume parametrization of Fel is rather simple as we sepa-
material, thefreeenergyincludesthefollowingcontribu-
ratedouttheT =0KbindingenergyE ,whichcarries
tions, 0K
a stronger volume dependence (fitted by a Murnaghan
equationofstate64). Inthiswayanumericalerrorofless
F(V,T)=E (V)+Fel(V,T)+Fqh(V,T)
0K than 0.1 meV/atom is achieved at all temperatures and
+Fph-ph(V,T)+Fel-ph(V,T), (1) volumes. For further details, we refer to Ref. 22.
where the electronic (el), quasiharmonic (qh) and the
B. Vibrational contributions
anharmonic phonon-phonon coupling (ph-ph) consti-
tute the adiabatic approximation22, while the electron-
phononcoupling(el-ph)containsalsonon-adiabaticcon- In Eq. (1), the terms Fqh and Fph-ph together char-
tributions. The target isobaric heat capacity C is de- acterize the vibrational free energy. The quasiharmonic
P
rived from the free energy using the relation, contribution, Fqh(V,T), describes non-interacting, vol-
ume dependent phonons and can straightforwardly be
(cid:18)∂2F(V,T)(cid:19) calculated22,65. The above mentioned Debye approx-
C = T . (2) imation for the harmonic lattice vibrations yields the
P − ∂T2 V,P expression18
In the following the employed methods to calculated the Ch = 12π4kB (cid:18) T (cid:19)3 (6)
different contributions to Eq. (1) are briefly discussed. V 5 Θ
D
3
for the heat capacity, which is cubic in T. Furtherstudies38,39demonstratethatthecouplingaffects
To estimate the influence of electronic temperature on electronic and vibrational degrees of freedom simultane-
the phonon frequencies, we have calculated the varia- ously.
tion of the highest optical phonon frequency for Al Sc
3
(marked by an orange arrow at point R in Fig. 5c) with
theFermismearingwidthσ rangingfrom0.01eVto0.15 D. Computational Details
eV (120 K – 1740 K). The variation of this phonon fre-
quency is on the order of 2.4 10 2 THz, which cor-
× − Total energy and force calculations are performed for
responds to an energy change of 0.1 meV. We have
≈ a 4 4 4 fcc supercell within the projector-augmented
determined the impact of the resulting temperature de- × ×
wave (PAW) method67 as implemented in the Vienna
pendenceontheheatcapacityandfoundittobeanorder Ab Initio Simulation Package (vasp)68,69. The gen-
of magnitude smaller than the contribution of Fqh.
eralized gradient approximation (GGA) as parameter-
In addition we have calculated the modification of the
ized by Perdew-Burke-Ernzerhof (PBE)70 is used for the
heat capacity due to explicitly anharmonic vibrations
exchange-correlationfunctional. InthecaseofAl,acom-
caused by phonon-phonon interaction. In accordance
parison with the local density approximation (LDA) has
with previous studies28, this effect also turns out to be
beendonetoestimatetheimpactwhichthechoiceofthe
negligibleforthetemperatureregimeinvestigatedinthis
exchange-correlation functional has on the results. As
work (<400 K). Having in mind that already the quasi-
discussed in previous studies11 these deviations can be
harmoniccontributionFqh isforthecriticaltemperature
used as an approximate DFT error bar.
regime (<5 K) much smaller than the electronic contri-
The phonon calculations are performed employing
bution Fel, we do not include the modification of lat-
the small displacement method22,71 with a displacement
tice vibrations due to electronic temperature or phonon-
valueof0.02Bohrradius( 0.01˚A).Thesmallvalueen-
phonon interaction in the upcoming discussion. ≈
sures that the forces on all the atoms within the given
supercell vary linearly with the displacement. After per-
forming convergence tests, a plane-wave energy cutoff
C. Electron-phonon coupling
E =400eVisused. TheMonkhorst-Packscheme72for
cut
the sampling of the Brillouin zone (BZ) is chosen, with
The electron-phonon coupling has been shown35 to be
a reciprocal-space k-mesh of 6 6 6 grid points. An
crucial at very low temperatures (typically 0.1–4 K) for energy of 10 7 eV is used as a×conv×ergence criterion for
−
describing the deviation of the electronic heat capacity
the self-consistent electronic loop. A Methfessel-Paxton
CPel from the linear temperature dependence according scheme73 with a width of 0.15 eV is used for the force
to Eq. (3) that is solely captured by the Sommerfeld co-
calculations entering the dynamical matrix.
efficientγ . Consideringthecoupling,theelectronicheat
The frequencies needed to calculate Fqh are generated
capacity is written as
bysamplingthecompleteBZusingaq-meshof30 30
× ×
30 grid points. A relative shift of (0.25,0.25,0.25) with
Cel+Cel-ph =γ(1+λ)T , (7)
P P respect to the Γ point is introduced. This is important
forthecalculationoftheheatcapacityinthelimitT 0
where λ is the dimensionless electron-phonon coupling
→
K,whereonlythemodesinthevicinityoftheΓpointare
parameter. This parameter is formally defined as
excited,whiletheΓpointitselfneedstobeexcludedfrom
the first reciprocal moment of the Eliashberg spectral
function34–37 α2g(ν), thepartitionsumduetoitsdivergingcontribution. Even
with the shift an artifical dependence on the choice of q-
(cid:90) ∞ α2g(ν) mesh employed for sampling the BZ is expected, which
λ=2 dν . (8)
makes the temperature region (T 0) unreliable. We
ν
0 →
findthatwitha4 4 4fccsupercellandaninterpolation
× ×
The Eliashberg spectral function represents the vibra- meshof30 30 30gridpointswiththe(0.25,0.25,0.25)
× ×
tional density of states (VDOS) g(ν) weighted by the ef- shift, the Debye behavior is valid down to 2 K. By in-
fective electron-phonon coupling function averaged over creasing the mesh size, for example, to 50 50 50 one
the Fermi surface α2 as explained in Refs. 34,37,66. can extend the Debye behavior to below×1.8 K×, but it
Physically, it quantifies the contribution of phonons would add to the computation time dramatically. Since
withfrequencyν tothescatteringprocessofelectronsat deviations of the experimental C /T3 data from linear-
P
the Fermi level34. The energy changes due to this cou- ity are already observed in the region between 2 K and
pling can be obtained from diagrammatic perturbation 20K(grayshadedregionsinFigs.3b), thelowerlimitof
theory. The resulting temperature dependence is mainly 2 K is sufficient for the purpose of the present study.
determinedbyaproductofFermidistributionandBose- For the calculations of the electronic free energy, a
Einstein distribution functions. In the low temperature Fermi smearing with a width ranging from 8.6 10 5
−
×
limit, however, only the T independent zero point vibra- eV to 0.137 eV is used, corresponding to a temperature
tions are relevant33,66, yielding an overall linear T de- range of 1-1590 K (the upper limit corresponds to the
pendence of Cel-ph similar to the Sommerfeld expansion. melting temperature for Al Sc) with a mesh of 11 sepa-
P 3
4
rate temperature values. The calculations are performed E. Experimental Details
for a unit cell with a k-mesh of 20 20 20 grid points
× ×
and a plane-wave energy cutoff of 300 eV.
AnAl Scingotwaspreparedbyinductionmeltingand
3
casting into a copper mold from pure Al (99.999 wt.%)
and Sc (99.99 wt.%) under vacuum conditions in the re-
search group of M. Rettenmayr (Otto Schott Institute
10 10 ofMaterialsResearch,Friedrich-Schiller-UniversityJena,
T=5 K T=10 K
8 8 Germany). The ingot was re-melted several times and
homogenized at 1000 C for 17 hours.
6 6 ◦
4 4
)
B 2 2
k
3
− 0 0
0 0 4 8 12 16 20 24 0 4 8 12 16 20 24
1
×
el (P 20 T=15 K 20 T=20 K
C 16 16
12 12
8 8
4 4
0 0
0 4 8 12 16 20 24 0 4 8 12 16 20 24
K-mesh (N)
FIG. 1: Convergence of the k-mesh for the electronic con-
tribution to the heat capacity (Cel) in k for T = 5, 10, 15
P B
and20K.TheN onthex-axisdenotesthek-meshsize(N3).
The gray shaded area corresponds to a width of magnitude
1×10−3k (boundaryvaluesdependontemperature)within
B
which the converged Cel fluctuates.
P
FIG.2: (Coloronline)Orientationimagingmicroscopyusing
EBSDanalysisoftheAl Scsample. TherecordedEDXspec-
3
trumisshownasinsetanditrevealsthepresenceofsolelyAl
Figure 1 shows the convergence of the electronic con-
and Sc atoms.
tribution to the heat capacity Cel with respect to the
P
k-mesh at the electronic temperature T = 5, 10, 15 and
Cube-shapedsamplesofsize3 3 3mm3 werecutby
20 K. For T = 15 K, for example, a mesh of at least × ×
sparkerosion, etched, andpolished. Low-temperature(2
12 12 12 k-points is required to ensure that the er-
× × to 400 K) heat capacity measurements were performed
ror bar of Cel is on the order of 0.001 k (gray shaded
P B byaPhysicalPropertyMeasurementSystem(PPMSTM,
regions in Fig. 1). Since the Cel values scale with T, one
P Quantum Design Inc.). The microstructure of the cast
needs at higher temperatures larger k-meshes to match
andannealedAl Scsamplewascheckedusingorientation
the same absolute error. For the present work, we chose 3
imaging microscopy via an electron back-scatter diffrac-
a mesh of 20 20 20 k-points.
× × tion(EBSD).Alargeareaof1500 1500µm2wasscanned
×
with a step of 2µm (Fig. 2). The points with the con-
The calculations to obtain the electron-phonon cou-
fidence index smaller than 0.1 were excluded. The sam-
pling parameter given by Eq. (8) are performed em-
ple composition and homogeneity were controlled by an
ployingdensityfunctionalperturbationtheory(DFPT)74
EDXanalysis (insetFig.2). Smallinclusions correspond
withPAWpotentials(notidenticaltothoseinVASP)as
to Al Sc particles and their volume fraction is estimated
implementedintheABINITcode75,76. Fortheexchange- 2
to be lower than 1%.
correlation potential, we used the LDA as parametrized
by Perdew and Wang77, since DFPT is in ABINIT cur-
rently not implemented for GGA. The chosen values for
III. RESULTS AND DISCUSSION
the energy cutoff and the k-point grid are 20 Ha ( 544
≈
eV) and 28 28 28, respectively. The calculations for
× ×
Al Scareperformedwiththeequlilibriumlatticeparam- In Figs. 3a and 3d, we present our measured (green
3
eter of 7.62 Bohr (4.032 ˚A). The LDA lattice constant crosses) and calculated (blue lines) C data for Al and
P
obtained with VASP is 4.033 ˚A, i.e. almost same as the Al Sc. We find a very good agreement over the whole
3
value for the ABINIT pseudopotential. The convergence temperature range. The good agreement with exper-
criterion for the self-consistent field cycle is 10 14 Ha on iment is consistent with the small deviations between
−
the wave function squared residual. LDA and GGA results, since these deviations have been
5
T (K)
1 2 3 4 5
3.5
Ck (/atom)PB123 22..89 EQQQQxhhhhp++ ((.EEGL (llDPG ((rAAGLes)D)GeAnAt))) 364/T ( 10 J/g.K)−×12..55 64( 10 J/g.K)−×012340 Silic50aT (K)QQQQQQ1hhhhhh0++++ ((0EEEEGLllllDG++ ((AccAGLoo)D)GuuAppAll))iinn1gg 5((GL0DGAA)) 2C/T (mJ/mol.K)P0112....50501111....40069910 EEW-Wxx-pp-i/ ..toW h(( PKee/lohlo-- ipkpelhll)hi- p pccsoho)u ucppolluiinnpggli - n((-GgG- --- GWG(---L---AA iDCCCt)h)A 111e)===l-p111h... c433o652u,,,pCCClin333g=== (L000D...A000323)
2.7 P
2.6 C 0.5 Exp. (Berg) Exp. (Present) Qh (GGA)
200 225 250 Exp. (Present) --- Qh (LDA)
0
0 100 200 300 400 3 10 50 100 0 5 10 15 20 25
Temperature (K) Temperature (K) T2 (K2)
(a)Al (b)Al (c)Al
T (K)
1 2 3 4 5
3.5
Ck (/atom)PB123 23..90 EQQxhhp+ (.EG (lPG (rAGes)GenAt)) 364T ( 10 J/g.K)−×12..55 64( 10 J/g.K)−×012340 Silica50 EQQQTxhhh p++ ((.KEEG (llPG+ )(rcAGeo1s)Geu0npAtl0))ing (GGA)150 2/T (mJ/mol.K)P112...050001...790264 Exp. (Present) --WW- C/ioth1 e e=l-l-pp1hh. c 0co4ouu,ppClilni3ng=g ( (G0GG.G0AA1))
/ C 0.5
2.8 P
C 0.5
2.7 Qh (GGA)
300 350 400
0
0 100 200 300 400 3 10 50 100 0 5 10 15 20 25
Temperature (K) Temperature (K) T2 (K2)
(d)Al3Sc (e)Al3Sc (f)Al3Sc
FIG. 3: (Color online) Isobaric heat capacity (C ) plots for Al and Al Sc. (a) and (d) Measured (× symbols) and calculated
P 3
(solidlines: GGA,dashedlines: LDA)C forAlandAl Scink /atomrespectively. ThelabelsQhandElcorrespondtothe
P 3 B
quasiharmonicandelectroniccontributions,respectively. Theinsetprovidesazoomintohighlighttheelectroniccontribution.
(b) and (e) Renormalized C /T3 curves for Al and Al Sc in J/g·K4. The filled (cid:7) symbols are experimental data for Al taken
P 3
from Ref. 80. The inset shows C /T3 up to 150 K on a linear T axis. The magenta curve labeled Silica is the renormalized
P
C /T3 foramorphoussilicatakenfromRef.81forcomparisonwiththecurrentwork. (c)and(f)PlotsofC /T inmJ/mol·K2
P P
versusT2 toestimatethecoefficientsC andC forAlandAl Sc. Theuppertwinx-axisshowstheabsolutetemperatureinK.
1 3 3
Thesolid(dashed)blackandpurplelinescorrespondtotheGGA(LDA)calculationswithout(W/o)andwithelectron-phonon
(el-ph)couplingrespectively. Thefilled((cid:78)and(cid:4))symbolsaretheexperimentaldataforAltakenfromRefs.82,83respectively.
Thegray,greenandorangedashedlinesarethelinearfitstotheavailablelowtemperatureexperimentsrepresentedbysymbols
inthesamecolor. Thecolorednumberswrittennexttothey-axesarethey-intercepts(C )correspondingtodifferentcalculated
1
curves. The coefficients C and C obtained from the fit to the available experiments are written in respective colors.
1 3
empirically found11 to provide for this quantity a confi- creases with temperature T, one would expect that it
dence interval for the predictive power (or accuracy) of becomes even less significant at lower temperatures.
the DFT calculations.
The DFT calculations allow by construction a separa-
A. Significance of the electronic contributions
tion into the various free energy contributions, whereas
the experimental analysis gives only access to the total
response of the material. Specifically, the red curves in DuetotheT3 decreaseofthevibrationalheatcapacity
Figs.3aand3dcorrespondtothevibrationalpartasde- towards low temperatures, the standard representation
scribed within the quasiharmonic (Qh) approximation. chosenintheplotsinFigs.3aand3disnotthemostsuit-
The blue curves (Qh+El) show the impact of the elec- ablewaytovisualizeanddiscussfeaturesinthelowtem-
tronic contribution which is small and of the same order perature (<50 K) regime. To analyze the validity of the
asthetinydifferencebetweentheoryandexperiment(see Debye approximation, the heat capacity is often (e.g., in
inset in Fig. 3a). A similarly small electronic effect has the literature for thermoelectric materials) renormalized
been reported in previous DFT studies for Al10,22,28 and with T 3. A perfect Debye dependence (Eq. 6) would
−
Alalloys78. Sincetheelectroniccontributionlinearlyde- then result in a constant temperature independent be-
6
T (K)
10 20 30 40 50 60
100 phonons will not change this behaviour. The same ap-
Exp. (Present) pliestoanharmoniclatticevibrations, sinceourprevious
calculations28 for Al and our more recent results79 for
80 Linear fit
) Al Sc only show a noticeable impact to the heat capac-
2 3
K
ity close to the melting point.
ol. 60
m The non-zero intercept with the y-axis in Figs. 3c and
J/ 3f, which determines C1 in (10), therefore, arises from
m
( 40 30 contributions linear in T. These are the electronic con-
T tribution and the electron-phonon coupling (7). We will
/ 20
P first focus on the electronic contribution. The values of
C
20 10 thecoefficientC obtainedontheonehandbyfittingthe
1
0 experimental data to Eq. (10) and on the other hand by
0 200 400 600 800 1000
computing the electronic contribution to the heat capac-
0
0 1000 2000 3000 4000 ity are 1.04 and 0.72 mJ mol 1K 2 for Al Sc and 1.32-
− − 3
T2 (K2) 1.46 and 1.09 mJ mol−1K−2 for Al, respectively. The
spread in the experimental data for the C value of Al
1
reflects the scatter in the available literature82,83. The
FIG. 4: (Color online) (Main) Linear fit (red dashed line)
sizeabledifferencebetweentheexperimentalandtheoret-
to the present experiments in the representation given by
icalC indicatesthatelectron-phononcouplinginEq.(7)
Eq.(10)uptoapprox.60K.Theupperx-axisrepresentsthe 1
cannotbeneglected. Thus,C isnotonlydeterminedby
absolutetemperatureT. (Inset)Zoomofthelowtemperature 1
region up to T ≈32 K (T2 =1000 K). the Sommerfeld coefficient γ, but also by the coupling
parameter λ. For Al (Fig. 3c), we have therefore com-
putedtheelectron-phononcouplingandhavedetermined
havior. Figures3band3eshowthatthevibrationalpart thevalue0.47forλ,ingoodagreementwiththereported
(red curves) exhibits only a weak temperature depen- λ values for Al that range from 0.38 to 0.4537,84,85. Us-
dence. ThesmalldeviationsfromCP T3 arediscussed ing our λ value in Eq. (7) we get C1 = 1.6 mJ mol−1
in the next section. For reasons give∝n in Sec. IID, the K−2 . This result agrees well with the experimental val-
heat capacityis notplotted below2 K.Wenote thatthe ues (1.32-1.46 mJ mol−1 K−2)82,83. In the case of Al3Sc
experimental measurements do not follow the constant (Fig. 3f) the calculation yields λ=0.33 (for Al3Sc there
behavior below T 20 K, but rather increase sharply, existsnopublisheddata)correspondingtoC1 =0.96mJ
therewith indicatin≈g the importance of non-vibrational, mol−1 K−2 and showing again a good agreement with
i.e. electronic excitations. the experimental value of 1.04 mJ mol−1K−2.
For this reason, experimental measurements are com- We can now revisit Figs. 3b and 3e. Taking in addi-
monly fitted to the sum of two contributions to describe tion to the vibrational contribution (red lines) also the
the temperature dependence of the heat capacity in the electronic contribution (blue curves) into consideration,
low temperature regime18,19, we observe that the ab initio determined heat capacity
agrees qualitatively as well as quantitatively well with
CP(T)=C1T +C3T3. (9) the experiments. Based on the insights obtained with
Figs. 3c and 3f, we can now conclude that the experi-
Thestructureofthisexpressionismotivatedbythecom-
mental data in the C /T3 plot below 5 K corresponds
P
petition of electronic and vibrational contributions, as
to a 1/T2 behaviour, i.e. C T. This is to a large
P
captured by the Sommerfeld model (3) and the Debye ∼
extend captured by the electronic contribution to C .
P
model (6), respectively. A straightforward approach to
Taking in addition the electron-phonon coupling into ac-
test the validity of Eq. (9) is to perform another renor-
count(brownlines),furtherimprovestheagreementwith
malization of the heat capacity data, namely by plotting
experiment. The latter contribution affects C typically
P
C /T =C +C T2, (10) in the temperature range between 0.1-4 K35,38. Due to
P 1 3 theobservedlinearityintheC /T versusT2 representa-
P
versus T2. As shown in Fig. 4, the experimental data tion in Fig. 4, however, we consider a temperature range
show in this representation a constant slope up to ap- up to 20 K for the electron-phonon coupling in Figs. 3b
prox.20K,i.e.,uptothistemperaturetheheatcapacity and 3e.
is exclusively captured by a T and T3 dependence. As mentioned before, the fcc crystal structure is com-
An advantage of the ab initio methodology is that it mon to both Al and Al Sc phases where the Sc atoms
3
allowsaseparatediscussionofthedifferentcontributions replace the Al atoms on the corner sublattice sites re-
to the heat capacity as shown in Figs. 3c and 3f. If only sulting in the L1 Al Sc phase. Comparing the above
2 3
lattice vibrations contributed to the measured heat ca- discussed features for Al and Al Sc (Fig. 3: all subplots
3
pacity at these temperatures (red lines), C /T would are provided on the same scales), we observe that the
P
passthroughtheorigin. AsalreadyindicatedinSec.IIB impact of the electronic contribution and the electron-
also the impact of a finite electronic temperature on the phononcouplingstrengthisreducedcomparedtothatof
7
g(⌫)/⌫2 g(⌫) g(⌫) g(⌫)/⌫2
(⇥10�2THz�3) (⇥10�1 THz�1) (⇥10�1 THz�1) (⇥10�2THz�3)
(a) Al (b) Al (c) Al + Al Sc (d) Al Sc (e) Al Sc
3 3 3
FIG. 5: (Color online) Calculated phonon dispersion curves of Al (solid red lines) and Al Sc (dashed blue lines) (c) plotted
3
alongapathintheBZforAl Sc,(bandd)thecorrespondingvibrationaldensityofstatesg(ν)(VDOS)and(aande)reduced
3
VDOS[g(ν)/ν2]forAl(left)andAl Sc(right). ThebrowncurverepresentstheDebyeVDOS(g(ν)∝ν2)uptotherespective
3
Debyefrequencies. Themagentacurve(takenfromRef.86)representsthereducedVDOSforamorphoussilicaforcomparison.
ThelabelsP ,P ,P ,P marktheobservedpeaksintheVDOS.TheorangearrowatpointRin(c)markstheselectedphonon
1 2 3 4
mode to study the effect of electronic temperature on the phonons (c.f. Sec. IIB).
pureAl. Thisisincontrasttoestimatesintheliterature, states (VDOS). Such maxima in C /T3 are extensively
P
accordingtowhichthecouplingconstantλofanalloycan discussed in the amorphous materials community81,89–92
be approximated by the concentration-weighted average (e.g., for silica) and are called boson peak. In contrast to
of the coupling constants of the individual elements39,87 the maxima we observe here, the peaks found for amor-
(compare λ = 0.33 for Al Sc with 0.47 for Al and 0.68 phousmaterialsaremuchsharper(magentacurvewitha
3
forSc88). ThereasonwhythisapproachfailsforAl Scis maximaataround15KininsetsofFigs.3band3e)and
3
the reduced density of states at the Fermi level in Al Sc are associated with localization phenomena (e.g., short
3
comparedtothatofthepuremetalsandinparticularAl. range order) that shift the van Hove singularities to the
This reduces the number of partially occupied electronic low-frequency regime (close to 1 THz).
states and thus the electronic entropy. The reducedden- In the crystalline and perfectly ordered materials in-
sity is the origin and driving force behind the energetic vestigated by us, the non-Debye behaviour in Figs. 3b
stability of this intermetallic compound, i.e., the fail- and 3e has a different physical origin93–95. To show
ing of the averaging approach is a direct consequence of this,wefirstprovidethephononspectraofAlandAl Sc
3
the fact that the electronic structure of the alloy cannot (Fig. 5c), plotted in the same BZ belonging to the L1
2
be regarded as small perturbation compared to the pure phase. The spectra are used to calculate the VDOS of
compounds. Since the same argument is expected to ap- Al and Al Sc (Figs. 5b and 5d). In addition, we pro-
3
ply to any intermetallic compound, the average concept vide the Debye VDOS96 (brown solid curves in Figs. 5b
developed for alloys39,87 cannot be extended to ordered and 5d) as g (ν) = 3ν2/ν3 with the Debye frequency
D D
compounds. νD =2π(9ρat/4π)1/3(2c−t 3+c−l 3)−1/3. Here,ctandclare
the transverse and the longitudinal velocities of sound
for metals calculated from the corresponding acoustic
B. Non-Debye behaviour in the heat capacity phonon branches close to the Γ point and ρ is the
at
atomicconcentration. Comparingthecalculatedandthe
In addition to features below 20 K, we observe in Debye VDOS (in Figs. 5b and 5d), we notice two pro-
the renormalized heat capacity (C /T3) plots (insets in found peaks (P and P ) between 4-6 THz as well as
P 1 2
Figs. 3b and 3e) maxima at approx. 40 K for Al and 50 additional peaks at higher frequencies that are charac-
KforAl Sc. Atthesetemperaturestheelectroniccontri- terized by an excess DOS over the Debye value. At the
3
bution is already negligible and the Debye model would same time, Figs. 5a and 5e compare the reduced VDOS
result in a constant line, i.e., the origin of these max- (g(ν)/ν2) of Al and Al Sc with that of silica for which
3
ima must be related to an excess vibrational density of we notice a sharp profound peak close to 1 THz.
8
It is difficult to experimentally resolve the effect the 2.0 2.0
peaks have onto the shape of the C /T3 curves, since Experiments-Al
P
the features are not very pronounced. In a theoretical All frequencies-Al
investigation, we can analyze the heat capacity after ex- 4K 1.5 Without 4-6 THz-Al 1.5 VC
tracting the frequency band of 4-6 THz. We use a repre- m ν=P1 Experiments-Al3Sc /T
sttoohefnitlstahaptettuiiBocrpenZo.mosAfeo,stdhiuinlellauoBtsritZdorenabrstyetfido6tt4aiincnehgFxieaiignvc.teto6aqa(wpd4aoe×lsilnh-4tde×sdefi(4ocnrsoeaurdnrpegweserpecicoegunlhlrd)tviifenno)ggr, 36T ( 10) J/−×1.0 ν=P2 EinstAWeililnt hf roCeuqVtu 4 e-→n6c iTeHs-zA-lA3Sl3cSc 1.0× ( 10) J/m−36
the procedure suppresses the maximum in CP/T3 for Al /P0.5 0.5 K
atT 40Kandcreatesanewrelativelyflatmaximumat C 4
aslig≈htlyhighertemperature. Flatmodesinthephonon ν=P3
ν=P
dispersion curve at higher frequencies (e.g., P3) are re- 0.0 4 0.0
sponsible for the remaining maximum. 0 50 100 150 200
To see how the addition of Sc affects the non-Debye Temperature (K)
behaviour, we compare the phonon dispersions of pure
Al with Al Sc (Fig. 5c): The phonon branches in Al Sc FIG. 6: (Color online) Effect of excluding the 4-6 THz fre-
3 3
are higher in frequency than in Al. This is attributed quencybandonthelowtemperaturepeak(orangeandgreen
to the stiffer nature of the Al-Sc bonds as compared to dashedcurves)intherenormalizedheatcapacityCV/T3 plot
(right y-axis) for Al and Al Sc. The solid orange and green
Al-AlbondsduetoastronghybridizationbetweenScd 3
− curves correspond to C /T3 considering all the frequency
and Al p electron states8 and also reflected by a higher V
− modes in the phonon dispersion. The fading gray curves are
bulk modulus value of 85.2 GPa for Al Sc in contrast
3 the Einstein C /T3 calculated at frequency ν correspond-
V
to 74.6 GPa for Al. In addition, we observe a different
ing to the peaks P ,P ,P and P in the VDOS of Al Sc
1 2 3 4 3
relevance of flat optical branches in particular above 6 (Fig. 5d) respectively. The dashed grey curve joins the max-
THz,yieldingthepeaksP3 andP4 intheVDOSofAl3Sc ima (grey circles) of the Einstein CV/T3 curves. Note that
(Fig. 5d). To check the impact of these peaks onto the theory is given as C /T3, whereas experimental data are
V
maxima in the heat capacity of Al Sc, we repeated the given as C /T3, resulting in an insignificant difference.
3 P
procedure of neglecting the frequency band of 4-6 THz
(which lead to P and P ). As shown in Fig. 6 (dashed
1 2
green curve), we again observe a significant reduction in mined by the low-lying flat regions in the phonon spec-
the maximum with a shift towards higher temperature trum, i.e. the region of 4-6 THz in Al and Al Sc is re-
3
as was also observed in the case of Al. Hence, the peak sponsible for the maxima around 40-50 K in the heat
P is less relevant than P and P . capacity. ComparingtheheatcapacitiesofAlandAl Sc
3 1 2 3
To evaluate the importance of these peaks in the (Figs. 3b and 3e), one can notice that the maximum is
VDOS, we use the Einstein model in which each phonon less pronounced for Al Sc and occurs at higher tempera-
3
branch is effectively replaced by a constant single fre- ture.
quencyν andtheVDOSexhibitsasinglediscretepeak.
E
The Einstein heat capacity is given by19,
IV. CONCLUSIONS
ekhBνET (hνE)2
C =3Nk (11)
V B(cid:26) hνE (cid:27)2(kBT)2 Based on the insights gained in this study, we demon-
ekBT 1 strate: (i)Thepredictivecapabilityoffinite-temperature
−
DFT in accurately describing the competition between
where N,k ,h are the number of atoms, the Boltzmann electronic and vibrational heat capacity contributions in
B
and the Planck constant, respectively. The critical tem- AlandAl ScattemperatureswellbelowtheDebyetem-
3
peratureatwhichtheheatcapacityhasalocalmaximum perature, if highly converged calculations are performed.
as per the Einstein model is T (ν) = hν /4.928k . In (ii) The importance of contributions beyond the adi-
E E B
Fig. 6 (fading gray curves), we plot the Einstein C /T3 abatic approximation, in particular of electron-phonon
V
curves for the frequencies corresponding to P ,P ,P coupling for the correct determination of the electronic
1 2 3
and P of Al Sc (Fig. 5d). With increasing frequency heat capacity at low temperatures. (iii) The connection
4 3
(ν < ν < ν < ν ), the maximum is observed to offlatbranchesinthephononspectrumandtheobserved
P1 P2 P3 P4
flatten and to shift towards higher temperatures. This maxima in the low-temperature heat capacity measure-
trend agrees with our foregoing comparison of maxima ments that are not captured by the Debye model.
for Al and Al Sc: The peaks in the Al Sc VDOS are at TheavailabilityofhighlyaccurateDFTcomputedheat
3 3
higher frequencies (owing to stiffer phonons) than those capacities including the partition in the various entropic
for Al. Consequently, the maximum is flatter and ob- contributionsallowedadetailedanalysisaboutthevalid-
served at a slightly higher temperature than Al. There- ity of approximate but commonly applied physical mod-
fore,thenon-Debyebehaviourofcrystalsismainlydeter- els such as Sommerfeld theory or Debye model. To visu-
9
alize and discuss the relevant low temperature features analysisofthedataaregeneralandcanbedirectlytrans-
we considered various renormalized transformations. ferred to other materials and compounds.
The comparison of Al and Al Sc reveals that the ob-
3
served deviations from the Debye behaviour at low tem-
peratures are particularly strong in the pure element Al. Acknowledgments
The addition of 25 at.% Sc to Al reduces the fcc sym-
metry due to the L12 superstructure. Despite the higher TheauthorsthankDr.S.LippmannandProf.M.Ret-
structuralcomplexity,however,thelow-temperaturefea- tenmayr (Otto Schott Institute of Materials Research,
tures in the heat capacity become less pronounced. This Friedrich-Schiller-University Jena, Germany) for fruitful
is related to the higher stiffness of the Al-Sc bonds and discussions and support in preparation of Al Sc sam-
3
the resulting increase of the Debye temperature. In an ples. TheauthorsarealsogratefultoProf.M.W.Finnis
Al-Scalloythatphaseseparatesintoasolidsolutionand and Dr. X. Zhang for valuable discussions and sugges-
Al3Sc precipitates, the thermodynamic properties at low tions. Financial support from the Deutsche Forschungs-
temperatureswillthereforebedeterminedbytheAl-rich gemeinschaft (DFG) within the priority program SPP-
matrix. While this study focussed on two example sys- 1713 “chemomechanics” (research projects HI 1300/8-1
tems – fcc Al and Al3Sc – all discussions and consider- and DI 1419/7-1) is gratefully acknowledged.
ations regarding the methodological approaches and the
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