Physically founded phonon dispersions of few-layer materials, and the case of borophene

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by  Jesús Carrete

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Physically founded phonon dispersions of few-layer materials, and the case of
borophene
Jesu´s Carrete,1 Wu Li,1 Lucas Lindsay,2 David A. Broido,3 Luis J. Gallego,4 and Natalio Mingo1,∗
1CEA, LITEN, 17 Rue des Martyrs, 38054 Grenoble, France
2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA
4Departamento de F´ısica de la Materia Condensada, Facultad de F´ısica,
Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain
An increasing number of theoretical calculations on few-layer materials have been reporting a
non-zero sound velocity for all three acoustic phonon modes. In contrast with these reports, here
we show that the lowest phonon dispersion branch of atomistically described few-layer materials
shouldbequadratic,andthiscanhavedramaticconsequenciesoncalculatedproperties,suchasthe
thermal conductivity. By reformulating the interatomic force constants (IFC) in terms of internal
coordinates, we find that a delicate balance between the IFCs is responsible for this quadraticity.
This balance is hard to obtain in ab-initio calculations even if all the symmetries are numerically
enforced a posteriori, but it arises naturally in our approach. We demonstrate the phenomenon in
the case of borophene, where a very subtle correction to the ab-initio IFCs yields the physically
correct quadratic dispersion, while leaving the rest of the spectrum virtually unmodified. Such
quadraticityneverthelesshasamajoreffectonthecomputedlatticethermalconductivity,whichin
thecaseofborophenechangesbymorethanafactor2,andreversesitsanisotropy,whenthesubtle
IFC correction is put in place.
I. INTRODUCTION elasticity theory predicts16. Indeed, many recent ab-
initio calculations report three linear-dispersionacoustic
branches (silicene17–19, phosphorene20, MoS 21, WS 22,
Exfoliated single-layer and few-layer materials have 2 2
MgB 23, WSe 24) whereas some other ab-initio calcula-
recently emerged as potentially revolutionary for many 6 2
different applications1. Following the first isolation of tions,andvirtuallyeveryempiricalpotentialcalculation,
report one quadratic and two linear acoustic branches
graphene, the number of exfoliated 2D materials to be
(silicene25, phosphorene26,27, MoS 28, WS 28). Argu-
studied continues to increase every year. Some recent 2 2
examples drawing large audiences are2 MoS and black ments have even been given to suggest that in a buckled
2
phosphorene3,4,whichareparticularlypromisingforelec- system the flexural branch dispersion should no longer
be quadratic19.
tronics and optoelectronics. Other few-layer materials,
such as silicene, are still just at the theoretical model- In order to clarify the situation, in this paper we
ing stage, but large experimental efforts are currently formulate a necessary condition for the force constants
underway to bring them into real applications.5,6 An ex- of any relaxed mechanical system to be physically cor-
citingdevelopmenthasjusttakenplacewiththefirstex- rect. This formulation automatically guarantees the ful-
perimental synthesis of borophene,7 whose existence had fillmentoftheconstraintsduetocrystalsymmetry,trans-
been theoretically predicted not long before.8–10 lational invariance and rotational invariance29–31. Using
thisapproachweshowthataquadraticdispersionacous-
The 2D character of these systems brings in unique
tic branch is always present on suspended few-layer sys-
physical properties, sometimes resulting in desirable
tems. Wealsoshowthatthiscanhaveastrongimpactin
qualities,suchashighelectronmobilities,orhighthermal
conductivities11,12. Acentralroleindeterminingmanyof the calculated thermal conductivity of some 2D systems,
and we illustrate it in the case of the Pmmn borophene
these properties is played by the phonon dispersion. An-
structure,arguablyoneofthemostcomplex2Dmaterials
harmonic phonon scattering, and electron-phonon cou-
studied ab-initio to date.
pling, fundamentally affect the thermal and electrical
conductivities of the material, and their magnitudes are
directly related to the features of the phonon dispersion,
particularlythelowerfrequencyspectrum. Theexistence II. THEORY
ornotofaquadraticacousticphononbranchinthespec-
trum can completely change the physics of the problem Ourstartingpremiseis: Thetotalpotentialenergyofa
at hand. It can for example lead to appreciable changes physical system of particles, in the absence of any exter-
in the thermal conductivity, which has been an object of nal fields, can always be expressed as a function of a set
debate in the particular case of graphene13–15. ofinternalcoordinatesthatarescalars. Suchcoordinates
Itisthereforesurprisingthatnoconsensusexistsasto can include interatomic distances, angles, dihedrals, and
whether unstrained few-layer systems should always dis- so on. This principle is a consequence of the fundamen-
play one flexural phonon branch with quadratic phonon tal symmetries of physics. It is also intuitively evident,
dispersion at long wavelengths, even though this is what as the internal coordinates fully determine the positions
3
of atoms relative to each other, while guaranteeing the The approach described allows for the random gener-
isotropyandhomogeneityofspace. Thisprincipleisfully ation of arbitrary atomic positions and force constants,
general and not related to any particular form of the en- whichsatisfytherelevantsymmetries“byconstruction”.
ergy itself. Asimpleillustrativeexampleisthatofa2Dinfinitechain
If the system is close enough to a point of static me- with motions restricted to the (x,y) plane, and 2 atoms
chanical equilibrium, its energy can be expanded around in the unit cell, and interactions to 3rd nearest neigh-
the equilibrium positions: E = (cid:80) ∂2E ∆x ∆x , bors. The relative position of the two atoms is random,
ij ∂xi∂x i j accountingfor2arbitaryparameters. Another5random
where the {x } are Cartesian coordinates. Since E de-
i
parameterscorrespondtotheW betweentheatomsin
pends only on the internal coordinates, l , we can write: l,l(cid:48)
n
two adjacent unit cells. We can then numerically com-
pute the phonon dispersion near Γ (q = 0), and evalu-
∂2E (cid:88) ∂l(cid:48) ∂2E ∂l ate the exponent n of the frequency dependence on wave
= . (1)
∂x ∂x ∂x ∂l∂l(cid:48)∂x number ω ∝ qn for the lowest acoustic branch. Within
i j j i
l,l(cid:48) 100000 random configurations, all of them had n com-
patible with 2, meaning that every system contains a
Sincethederivatives ∂l areknown,thisprovidesuswith
∂xi quadratic dispersion.
the most general method to generate physically correct
The analysis would not be complete, however, unless
interatomicforceconstants. Itsufficestofirstchoosethe
we can address the phonon dispersion of real few-layer
equilibrium positions, from them compute all the inter-
systems calculated from first principles, and ascertain
nal coordinate derivatives, and then assign values to the
whether they always contain a quadratic acoustic mode.
energyderivativeswithrespecttoallpairsofinternalco-
The approach is to find the set of W that, using Eq.
ordinates. To guarantee that the system is dynamically l,l(cid:48)
(2), produce a set of physical IFCs that is closest to the
stable, the ∂2E needs to be symmetric and positive def-
∂l∂l(cid:48) raw (i.e. unphysical) ab-initio calculated IFCs. As we
inite. This can be ensured by generating it as the square
will see, in every case studied just very small changes of
of a matrix: ∂∂l2∂El(cid:48) =(cid:80)kATlkAkl(cid:48). some IFCs are able to turn the set into a physical one,
Such a representation is overcomplete, however: dif- resulting in dispersions that virtually do not differ from
ferent choices of the ∂2E may yield the same force con- the raw ones, except for the perfect quadraticity of the
∂l∂l(cid:48)
stants. This is easily resolved by the fact that all inter- lowest acoustic mode.
atomic distances can be expressed as sole functions of a The procedure to find the closest physical set of IFCs
minimal set of d×n internal coordinates. For n atoms for a given equilibrium system is as follows. First of
in 3D, it suffices to choose three atoms and specify their all, the raw IFCs, Hαβ are calculated, between the α
0,j
threeinternalcoordinatesplusthe3(n−3)internalcoor- and β degrees of freedom of the central unit cell and its
dinates relating them to the remaining atoms, resulting jth neighbouring unit cell, respectively. The standard
in3n−6values. Similarlyinstrict2Dweneedtochoose enforcement of translational invariance ensures that the
two reference atoms, and consider only 2n − 3 values. three acoustic phonon modes pass through zero at the
This has an important consequence: in 3D (2D) 6 (3) Γ point ((cid:126)q = (0,0,0).) However, in general, the three
of the system’s vibrational eigenvalues are always 0, and modes have a linear component, as shown in Fig. 1.
correspond to the rigid translations and rotations of the
Then we define a suitable set of internal coordinates
system.
for the extended system. We only need to consider the
Thusformulated,theapproachisvalidforatomicclus-
atoms in the central unit cell, and those interacting with
ters or molecules. To apply it to periodic systems, we
them within neighboring cells. We thus pick 3 of those
shallconsider(asitisstandardpractice)thatthesystem
atoms, makingsuretheyarenotcollinear, whichwillact
is a periodic array of supercells which interact only with
as a base, and we take their three interatomic distances
those supercells adjacent to them. Thus, the energy can
as coordinates. We could in principle continue picking
be fully expressed as a sum over adjacent pairs of su-
one of the remaining atoms at a time, and selecting its 3
percells, p (with p also including the case of interactions
distancestoeachofthebasisatomsascoordinates. How-
within the same cell.) The force constants can then be
ever, this approach runs into problems if the new atom
written in the fashion previously described, as
liesinthesameplaneasthetriangledefinedbythethree
basis atoms. In such case, the internal coordinates thus
H =(cid:88)(cid:88) ∂l(cid:48) W(p) ∂l , (2) defined cannot describe out-of-plane motions. To avoid
i,j ∂x l,l(cid:48) ∂x
j i thisproblem,whenthenewatomiscoplanarwiththeba-
p l,l(cid:48)
siswechooseadifferentsetofcoordinatescorresponding
whereW(p) = ∂2E,forl,l(cid:48) ∈{p},andpdenotesthepair to
l,l(cid:48) ∂l∂l(cid:48)
of neighboring supercells. Since the system is periodic,
W(p) must be defined in such a way that it does not y1(j) =|(cid:126)xj −(cid:126)x1|, (3a)
l,l(cid:48)
change if the atoms involved in the definition of internal y(j) =d(cid:126) ·(d(cid:126) ×d(cid:126) ), (3b)
2 j,1 2,1 3,2
coordinates l and l(cid:48) are simultaneously translated by a
y(j) =d(cid:126) ·[(d(cid:126) ×d(cid:126) )×d(cid:126) ], (3c)
lattice vector. 3 j,1 3,2 2,1 3,2
4
where d(cid:126) = (cid:126)x −(cid:126)x . There are many possible choices back to 2007. It comes as little surprise, then, that the
i,j i j
other than the above one. From now on we will label study of the 2D allotropes of boron has also progressed
internal coordinates with a single index, as y , rather through iterative improvement upon previous results.
l
than two.
a)
The next step is to construct the matrix η relating
the IFCs in Cartesian and internal coordinate represen-
tations. In other words, we want to express the force
constants as
(cid:88)
H = η W . (4)
i,j (i,j),(l,l(cid:48)) (l,l(cid:48))
l,l(cid:48)
Thus,
η =(∇ y )⊗(∇ y ). (5)
(i,j),(l,l(cid:48)) (cid:126)xi l (cid:126)xj l(cid:48)
η is non zero when atoms i and j are involved in the
internal coordinates y and y . This means that η is
l l(cid:48)
a sparse object, which permits us to take advantage of
sparse matrix arithmetic in the calculation. However, it b)
isimportanttorealizethatthevalueofW(l,l(cid:48)) willaffect Acoustic-XY Acoustic-Z Optical
not only each pair of atoms (i,j) belonging to y and y , / / Physical / / Raw
l l(cid:48)
180
but also all those pairs equivalent to the first one by a
symmetry operation of the space group (Bravais lattice 160
translations and point group symmetries). Thus, in the 140
process of constructing η, these equivalent pairs˜i,˜j have 120
to be identified, and the values of η(˜i,˜j),(l,l(cid:48)) properly set eV) 100
alongside with those of η . m
The last step consists(i,ijn),(ls,ol(cid:48)l)ving Eq. (4) by least E ( 80
60
squares. Inpracticalterms,thiswasdonebyconsidering
40
H and W as sparse arrays, and η as a sparse matrix.
The newly obtained H is the closest one to the original 20
set that satisfies the conditions required for the system 0 25
to be physical. 20
15 V)
e
m
III. APPLICATION TO BOROPHENE PHONON 510 E (
DISPERSIONS
0
X S Y
Theresultoftheapproachjustdescribedisillustrated
for a novel quasi-two-dimensional material: borophene.
FIG. 1. (a) Atomic structure of Pmmn boron, showing 25
The possibility of a boron-based analogue of graphene – unit cells. All atoms belong to the same element; colors are
borophene–hadrecentlybeentheoreticallypredicted8–10 onlyintendedtoguidetheeye. (b)Comparisonbetweenraw
and its first experimental demonstration has been just (dashed) and physical (solid) phonon dispersions.
reported7. Boron is carbon’s neighbor in the periodic
table and has similar valence orbitals, but its electron
deficiency prevents it from forming graphene-like pla- Densityfunctionaltheory(DFT)calculations8 suggest
nar honeycomb structures. The study of boron and its that single-atomic-layer B sheets composed of triangular
compounds is singularly difficult, to the point of being and hexagonal motifs are locally stable, the most stable
considered by some authors as “arguably the most com- structure of this kind being called the α-sheet. Theo-
plex element in the periodic table”32. Many proposed retical and experimental studies on the quasi-planar B
36
new phases of boron have been revealed as being ac- cluster with a central hexagonal hole seem to support
tually boron-rich compounds with very complex crys- this view9. However, more recent calculations10 have
tal structures33. In fact, boron displays multiple bulk predicted two novel 2D boron structures with non-zero
phases33,34, and the formation of clusters, fullerene-type thickness that are considerably more stable than the α-
cages and nanotubes made of B is also a subject of on- sheet. These two structures represent a more radical de-
going debate35–42. The ambient-conditions stable phase parture from the graphene prototype than silicene, ger-
of B was only recently determined through a series of manene and their binary relatives. Nevertheless, one of
computational studies, the earliest of which43 only dates thesephases,belongingtospacegroupPmmn,alsoshows
5
a distorted Dirac cone in its electronic band structure. The same effect was obtained for all the systems we
Hence,thisphaseisextremelyinterestingasthefirstma- havetried. Anylinearityofthelowestbranchgoesaway,
terial with massless fermions that is not closely related whereas the rest of the dispersion remains virtually un-
to a graphene-like honeycomb structure10. changed. ThechangesinIFCsinvolvedarereallysubtle,
which is the reason why it is very difficult to get the cor-
Pmmn borophene has a buckled structure, containing
rect quadratic dependence of the lowest branch by sim-
8 atoms in the unit cell, and being considerably more
ply imposing the various symmetries numerically. The
complex than those of other 2D materials like graphene
“by construction” approach described, on the contrary,
or phosphorene for example (see Fig. 1). Therefore, it
is able to capture this subtle effect with high precision,
is particularly well suited to illustrate the existence of a
since the physical constraints on the IFCs are built-in.
quadraticphononbranchinfinitethickness2Dmaterials.
Details of the ab-initio calculations are provided in the It has been reported that at finite temperature, the
supplemental information. Our fully relaxed orthorhom- anharmonic coupling in a single 2D layer renormalizes
bic unit cell has side lengths a = 4.52˚A and b = 3.25˚A, the low frequency flexural branch in the long wavelength
almost identical to Zhou et al.’s result10. The difference limit to ωZA(q) ∝ qα, with values for α ranging from
between the maximum and minimum atomic positions about45 1.2 to46 3/2 having been proposed. We note
along the OZ axis is 2.21˚A. The slab is formed by four thatthisrenormalizatonhasnothingtodowiththefinite
atomic layers along this axis, with an average distance groupvelocityoftenobtainedbyab-initio calculationsof
between them of 0.74˚A. zero temperature force constants, which is definitely an
artifact. Any calculation of thermal conductivity should
The raw calculated phonon dispersion displays a clear
employ physically correct IFCs, which yield a quadratic
linear mode along the Γ-X direction. By “raw” we mean
dispersionforthelowestbranchintheharmonicapprox-
calculated in the conventional fashion, where the IFCs
imation. This branch could then be renormalized for
that result from the ab-initio calculation are only cor-
anharmonic effects if so desired.
rected by enforcement of translational invariance, as im-
plemented e.g. in Phonopy44. Application of the previ- It is also important to mention that thermal fluctu-
ations in 2D systems lead to ripples, and destroy long
ously described approach results in a negligible change
range order47. The magnitude of this rippling effect de-
of the dispersion, except for turning the lowest acoustic
pendsonthesystem,andevenonthegrowthconditions.
mode into a perfectly quadratic dispersion near Γ (Fig.
Nevertheless,forwavelengthslongerthanthetypicalcor-
1). Looking at Fig. 2, the change in IFCs is very small.
rugation distance, the speed of sound of one of the three
Despite this, the effect on the frequency dependence of
acoustic modes will still tend to zero, as predicted by
the lowest branch is dramatic.
elasticity theory.
An alternative procedure to the “by construction” ap-
102 proach proposed is to symmetrize the IFCs by enforcing
all the different invariance conditions afterwards. This
101 second approach seems to us more complicated, and less
suitable than the “by construction” method, for the fol-
lowing reasons. The invariance of the potential energy
100
with respect to translations, rotations, and crystal sym-
metry operations imposes constraints on the interatomic
2) 10-1 force constants. However, as pointed out previously48,
Å
V/ in typical supercell or DFPT force constant calculations,
e
(al 0 such constraints are in general not satisfied because of
hysic intrinsic errors in the computational algorithms (e.g.
Kp -10-1 fromincompletebasissetsorimperfectself-consistency).
Therefore, these invariance constraints must be explic-
-100 itly imposed on the force constants after they have al-
readybeencalculated. Inafullyrelaxedbulkcrystal,the
-101 harmonic force constants must satisfy the Huang invari-
anceconditions29,whichgives15independentconstraint
-102 equations taking account of the symmetry. In addition
-102 -101 -100 -10-1 0 10-1 100 101 102 to these, there are 9k equations each imposing transla-
K (eV/Å2) tionalandrotationalinvariance,wherekisthenumberof
raw
atoms ina primitiveunit cell. These18k+15conditions
mustbeimposed,ontopofthoseassociatedwiththespe-
FIG.2. Comparisonbetweenraw(ordinate)andphysical(ab- cific crystal symmetries. Various approaches to do this
scissa) IFC values. (Symmetric logarithmic axes with linear havebeenimplemented(χ2 minimization,Lagrangemul-
threshold at 10−1eV/˚A2.) tipliers, and so on and so forth), but the best way to do
this remains ambiguous (e.g. allowing only a maximum
6
percent change in IFCs vs. an absolute change). phonons that carry most of the heat, due to their longer
lifetimes. Therefore,theeffectonlatticethermalconduc-
tivity is substantial. We point out that the first experi-
IV. EFFECT OF CORRECTED DISPERSION mentallysinthesizedsamplesofboropheneshowmetallic
ON THERMAL CONDUCTIVITY characteristics,7 in which case an electronic contribution
would add up to the total thermal conductivity. This
metallic character might be linked to the fact that the
a)
samplesaregrownonasubstrate,ratherthansuspended.
250
a, raw
200 b, raw
K) a, physical V. CONCLUSIONS
W/
150 b, physical
9
-0 WehavereformulatedtheexpressionforIFCsinterms
1
W( 100 ofinternalcoordinates,thusensuringthattheyarephysi-
callyvalid. Inthisformulationthelowestphononbranch
50 of an atomistically described finite thickness layer is al-
ways quadratic in the harmonic approximation, in con-
0 trast with the linear dispersion sometimes found in ab-
200 300 400 500 600 700
initio calculated phonon spectra of suspended layers.
T(K)
This quadraticity is linked to very subtle differences in
b)
102 the values of the IFCs, and it is hard to come by even
Raw if translational and rotational symmetry conditions are
-1s) 101 Physical numerically enforced. On the contrary, in our formula-
(p 100 tiontheIFCsarephysicallyvalid“byconstruction”,and
grate 10-1 twhietyhoauuttaonmyatnieceadllytosaimtispfyosealtlhtehme irnedqiuviirdeudalslyym. mByetrniues-
n
eri 10-2 merically finding the physical IFC set that is closest to
att the ab-initio one, we obtained phonon dispersions that
c 10-3
S arealmostindistinguishablefromtheoriginalonestothe
0 bare eye, except for the fact that their lowest phonon
10-1 100 101 102 branch is quadratic at Γ. In the case of borophene this
E (meV) has a dramatic effect on the computed thermal conduc-
tivity,whichdecreasesby50%andreversesitsanisotropy
when the correct physical force constants are used. We
FIG.3. Comparisonbetweencalculatedthermalconductivity
conclude that the linear dispersion of the out-of-plane
(a)andscatteringrates(b)usingraw(red)orphysical(blue)
modes often observed in ab-initio calculated spectra of
IFCs. The thermal conductivity of 2D systems has units of
2D and few-layer materials is an artifact resulting from
W/K. In order to compare with bulk materials one can di-
unphysical IFCs; any calculation dealing with phonons
vide the 2D conductivity by a chosen thickness of the film.
in2Dmaterialsshouldensurethephysicalcorrectnessof
For single or very thin layers, such thickness is ambiguous,
however, so we have preferred to keep the 2D units here. the IFCs, as we have shown here.
The seemingly minor corrections of the phonon dis- ACKNOWLEDGMENTS
persion shown above can have a dramatic effect on
the computed lattice thermal conductivity. In the case This work has been partly supported by the Air Force
of borophene, the physically correct phonon dispersion Office of Sponsored Research under grant No. FA9550-
yields κ values which are less than half of those ob- 15-1-0187, the European Union’s Horizon 2020 research
tainedwiththeunphysicallineardispersionfromtheun- and innovation programme under grant agreement No.
corrected DFT calculation. Even more striking is the 645776 (ALMA), and the M-Era programme through
fact that the anisotropy is reversed: the correct result project ICETS. L.J.G. acknowledes the support pro-
is κ > κ , whereas the linear dispersion calculation vided by the Spanish Ministry of Science and Innova-
b a
wronglypredictsκ >κ . Thisisrelatedtothelargeen- tion (Grant No. FIS2012-33126) and by the Xunta de
a b
hancement of the low frequency phonon scattering rates, Galicia (AGRUP2015/11). L. L. acknowledges support
which is produced by the larger density of states of the from the U. S. Department of Energy, Office of Science,
quadratic branch. As shown in Fig. 3, the phonon scat- Office of Basic Energy Sciences, Materials Sciences and
tering rates are nearly unchanged, except for energies Engineering Division for work done at ORNL. D.A.B.
below 10meV, where the physical ones are roughly 10 acknowledgessupportfromtheNSFEFRI2-DAREpro-
times larger than the raw ones. It is these low frequency gram under grant No. 1402949 and from ONR under
7
grant No. N00014-13-1-0234.
∗ [email protected] 1093–1098.
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