Electronic Structure and Phase Stability of Yb-filled CoSb3

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Article

Electronic Structure and Phase Stability of Yb-filled CoSb Skutterudite Thermoelectrics from First Principles 3

Eric B Isaacs, and Christopher Wolverton Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.9b01630 • Publication Date (Web): 05 Aug 2019 Downloaded from pubs.acs.org on August 8, 2019

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Chemistry of Materials

Electronic Structure and Phase Stability of Yb-lled CoSb3 Skutterudite Thermoelectrics from First Principles Eric B. Isaacs and Christopher Wolverton

∗

Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA

E-mail: [email protected]

Abstract

strongly inuence the electronic structure. The emergent conduction bands are spatially local-

Filling the large voids in the crystal structure

ized in the Yb-rich regions, unlike the delo-

of the skutterudite CoSb3 with rattler atoms

calized electronic states at the Brillouin zone

R

center that form the unlled skutterudite band

provides an avenue for both increasing car-

rier concentration and disrupting lattice heat

edges.

transport, leading to impressive thermoelectric performance. While the inuence of

R

on the

Introduction

lattice dynamics of skutterudite materials has been well studied, the phase stability of

RIn thermoelectric heat-to-electricity conversion, 2 the gure of merit is ZT = σS T /κ, where

lled skutterudite materials and the inuence of the presence and ordering of

R

on the elec-

σ

tronic structure remain unclear. Here, focusing

is the electrical conductivity,

κ

S

is the ther-

mopower,

employ rst-principles methods to compute the

is the temperature. Therefore, ecient thermo-

phase stability and electronic structure.

Yb-

electric materials must exhibit a rare combina-

lled CoSb3 exhibits (1) a mild tendency for

tion of electronic and thermal transport prop-

phase separation into Yb-rich and Yb-poor re-

erties: large

gions and (2) a strong tendency for chemical

to (1) understand the ability of existing ther-

decomposition into CoSb and YbSb binaries

moelectric materials, typically heavily doped

(i.e., CoSb3 , CoSb2 , and YbSb2 ). We nd that,

semiconductors, to satisfy this set of rare physi-

at reasonable synthesis temperatures, congu-

cal properties and (2) design improved thermo-

rational entropy stabilizes single-phase solid so-

electric materials, a detailed understanding of

lutions with limited Yb solubility, in agreement

the electronic structure, lattice dynamics, and

with experiments.

phase stability is critically important.

Filling CoSb3 with Yb in-

is the thermal conductivity, and

T

on the Yb-lled skutterudite Ybx Co4 Sb12 , we

σ,

large

S,

and small

κ.

In order

creases the band gap, enhances the carrier ef-

One famous class of thermoelectric materials

fective masses, and generates new low-energy

is the skutterudite CoSb3 , a covalent semicon-

emergent conduction band minima, which is

ductor satisfying the 18-electron rule.

distinct from the traditional band convergence

whose skutterudite crystal structure is shown in

picture of aligning the energies of existing band

Fig. 1, can be considered a perovskite (ABX3 ,

3

CoSb3 ,

is neces-

with an empty A-site) with substantial distor-

sary to achieve the emergent conduction band

tions of the CoSb6 octahedra that create large

minima, though the rattler ordering does not

voids.

extrema.

The explicit presence of

R

1

CoSb3 has a body-centered-cubic (bcc)

ACS Paragon Plus Environment 1

Chemistry of Materials 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 17

lattice with 16 atoms in the primitive unit cell and a space group of

Im¯3.

CoSb3 -based

thermoelectric materials exhibit favorable electronic transport properties, as the highly covalent bonding leads to large electronic mobility

µ

and

σ

κe , the electronic κ). 4 In addition, the presence of

(but also increasing

contribution to

a high-degeneracy conduction band minimum close in energy to the conduction band minima at the Brillouin zone center has been invoked to rationalize the large convergence.

5,6

S,

via the concept of band

Perhaps the most distinguishing feature of skutterudite materials is their ability to host rattler atoms

R (such as alkali, alkaline earth,

actinide, rare earth, and halogen elements) in the large crystallographic voids,

7

which serves a

dual purpose with respect to thermoelectricity. 2 First, it enhances the power factor (σS ) via electronic doping. duces

κL ,

8

Secondly, it drastically re-

the lattice component of

κ. 9

Loosely

bonded to the rest of the solid, the rattler atoms are believed to disrupt phonon transport via rattling in the voids (hence the name).

1013

While the inuence of rattlers on the lattice dynamical properties of skutterudite materials has been much studied,

1438

the phase stability

and electronic properties have received far less attention.

In particular, the thermodynamic

stability of lled skutterudite materials, the orFigure 1: Skutterudite crystal structure.

dering tendencies of the rattlers, and the pre-

The

cise inuence of the rattlers on the electronic

conventional unit cell is shown with the black

states are all unclear. Therefore, in this work,

dashed lines and cyan, blue, and brown cir-

we present a detailed study of the phase stabil-

cles indicate Yb/vacancy, Co, and Sb, respec-

ity and electronic structure of lled skutterudite

tively. We note that there is an alternative crys-

CoSb3 using rst-principles calculations.

tal structure description in terms of Sb4 rings (sometimes called squares or rectangles);

1,2

We

focus on Yb rattlers since Yb-lled skutteru-

one

dite CoSb3 exhibits some of the most promising

such ring is shown.

thermoelectric properties, e.g.,

ZT

approaching

1.5, and has been subject to considerable experimental investigation.

7,39

We nd that the Yb-lled skutterudite exhibits a tendency to phase separate into Yb-rich and Yb-poor regions, though the energetic lowering (compared to the completely empty and lled endmembers) is only on the order of 10 meV per Yb/void site. Due to the small magnitude of the formation energy, congurational entropy will likely win this energetic battle,


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Chemistry of Materials 51

consistent with the single-phase solid solutions

0.1 eV 1st-order Methfessel-Paxton smearing

typically found in experiment.

for structural relaxations, and the tetrahedron

The Yb-lled

52

skutterudite is in a three-phase region of the

method with Blöchl corrections

thermodynamic convex hull, with a substantial thermodynamic driving force for chemical de-

runs. The energy and ionic forces are converged −6 −2 to 10 eV energy and 10 eV/Å, respectively.

composition into binaries.

Given the reaction energy for Yb + 2CoSb3

We nd that this

for static

→

chemical decomposition tendency limits the Yb

YbSb2 + 2CoSb2 changes by only 20 meV/Yb

solubility, in agreement with experiments. Fill-

(1.3%) via the inclusion of the

ing the CoSb3 skutterudite with Yb opens the

the Yb PAW potential, we expect the absence

electronic band gap, increases the carrier eec-

of such states will not signicantly aect our

tive masses, and leads to the emergence of sev-

results.

eral new conduction band minima. The explicit

4f

states in

The convex hull is constructed from the Open

53,54

presence, though not the ordering, of the rat-

Quantum Materials Database (OQMD),

tlers is responsible for the new conduction band

database of electronic structure calculations

minima, which are not present in the unlled

based on DFT, which contains 49 binary and

skutterudite, as would be the case for the tra-

5 ternary phases in the YbCoSb space (as

ditional band convergence picture.

of June 2018).

The emer-

A cluster expansion (CE)

a

55,56

gent conduction bands exhibit distinct charac-

is employed to describe the energetics of dif-

ter with spatial localization in the Yb-rich re-

ferent congurations of Yb and

gions, as compared to the delocalized electronic

sublattice of voids in the skutterudite structure.

states at the Brillouin zone center.

The optimal CE in

on the bcc

atat 57 contains null, point,

and pair (out to 6th nearest neighbor) clusters. Disordered structures are modeled with special

Computational Methodology

quasirandom structures (SQS) sites for

As can be observed in Fig.

1, CoSb3 exhibits + + + a signicant octahedral distortion (a a a in Glazer notation

x

58

of 1/4, 1/2, and 3/4.

with 8 Yb/

59

an analytical representation of the

To generate

x-dependent

energetics of SQS, we t the formation ener-

40

) with respect to the ideal per◦ ovskite structure. While ∠CoSbCo is 180 for ◦ perovskite, it is 127 in CoSb3 . Similarly, ∠Sb ◦ ◦ CoSb is 8595 instead of the ideal 90 . The

gies of the SQS to a Redlich-Kister polynomials of order 1 (subregular solution model), as discussed in Refs. 60 and 61.

Band structure

unfolding based on the CoSb3 lattice parameter

octahedral distortion in the skutterudite crystal

is performed using

structure yields 1 void per 4 Co atoms. There-

bandup. 62,63

fore, the general stoichiometry for a lled CoSb3 skutterudite is Using

Rx Co4 Sb12

with

Results and Discussion

0 < x < 1.

to explicitly indicate an empty void,

Phase stability

1−x Rx Co4 Sb12 . We note limit of x, i.e., the lling frac-

the formula becomes that the upper

tion limit (FFL), is lower than 1 in practice,

4144

Figure

(DFT)

vasp

45,46

47

contains

the

YbCoSb

that Ybx Co4 Sb12 is in a 3-phase region of the

0 ≤ x ≤ 1.

Plane-wave

which

ternary convex hull based on the OQMD, shows

but we consider the full crystallographic range of

2(a),

density

functional

convex hull bounded by CoSb3 ,

theory

CoSb,

and

Yb11 Sb10 . In other words, Ybx Co4 Sb12 → (4 − 5x/11) CoSb3 + (5x/11)CoSb +(x/11)Yb11 Sb10

calculations are performed using

w/ the generalized gradient approxima-

48

tion of Perdew, Burke, and Ernzerhof using 6 2 Co, Sb, and Yb_2 (5p 6s valence) projec-

is the lowest-energy decomposition reaction ac-

tor augmented wave (PAW) potentials.

suggest the competing phases CoSb3 , CoSb2 ,

use a 500 eV kinetic energy cuto,

k -point

grids of density

≥

500

49,50

cording to the OQMD. Experiments instead

We

and YbSb2 ,

Γ-centered

k -points/Å

−3

42,64

so we focus on the correspond-

ing decomposition reaction.

,


Although CoSb2


(a)

(b)

Yb

Stable Phase

Page 4 of 17

Formation Energy

20

(meV/❒1−xYbxCo4Sb12)

10

Metastable Phase x=1 Composition

Yb5Sb3 Yb4Sb3

Competing phases in experiment Competing phases in the OQMD

CE Structures

Yb11Sb10

YbCo2

DFT Structures SQS

YbSb2

YbCo3

SQS polynomial fit

Yb2Co17 CoSb2 Co

CoSb

(110) (100) superlattice superlattice

−10 −20

b3 CoS

−30 −40

YbCo4Sb12 CoSb3

0

b3 CoS

0

❒Co4Sb12

Sb

+ oSb 2

+C

+ oSb

+C

1/4

YbS

b2

Sb 10 Yb 11

1/2

3/4

1

x in ❒1−xYbxCo4Sb12 YbCo4Sb12

Figure 2: (a) YbCoSb ternary convex hull from the OQMD. In addition to the stable phases (black lled circles connected by grey tie lines), we indicate two metastable phases (above the hull by 7 meV/atom for YbCo3 , 4 meV/atom for CoSb2 ) with open blue squares.

Ybx Co4 Sb12

corresponds to the line (not drawn) between the YbCo4 Sb12 composition (lled green triangle) and CoSb3 . The vertices of the red dashed triangle indicates the decomposition products for the lowest-energy decomposition reaction for Ybx Co4 Sb12 ; those of the smaller purple dotted triangle correspond to an alternate decomposition reaction discussed in the text.

(b) Cluster expansion

formation energy as a function of Yb concentration for structures used to t the cluster expansion (open blue circles) and those predicted by the cluster expansion (solid orange squares). The DFTcomputed formation energy for SQS (large open cyan circles) and a polynomial t (cyan line) are also shown. In panel (b), the thick lines in the region of negative formation energy correspond to the formation energy of the two decomposition reactions indicated in panel (a).


Page 5 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−kB T [(1 − x) ln(1 −

is metastable, it is above the convex hull by

to the mixing free energy,

only 4 meV/atom. Given this small energy, it is conceivable that vibrational entropy (not com-

x) + x ln(x)], where kB is the Boltzmann constant, is ≈ −44 meV for x = 1/4 at 900 K,

puted in this work) might stabilize this phase.

which is signicantly larger in magnitude than

Articially lowering the energy of CoSb2 alone

the mixing energy (≈

10

meV).

→ (4−2x)

The thermodynamic driving force for chem-

CoSb3 + 2x CoSb2 + x YbSb2 the lowest-energy

ical decomposition into binaries, on the other

decomposition reaction. However, as discussed

hand, is much stronger than the phase sepa-

in the Supporting Information, its combina-

ration tendency. As shown by the steep nega-

tion with stabilizing YbSb2 and/or destabiliz-

tively sloped lines in Fig. 2(b), the formation

ing Yb11 Sb10 can achieve this eect.

is not sucient to make Ybx Co4 Sb12

For ex-

energies for chemical decomposition are signi-

ample, the three-phase equilibrium of CoSb3 ,

cantly larger in magnitude than the mixing en-

CoSb2 , and YbSb2 is achieved by the simulta-

ergy.

neous articial energy lowering of CoSb2 and

the appropriate linear combination of CoSb3 ,

YbSb2 by 20 and 23 meV/atom, respectively.

CoSb2 , and YbSb2 [dotted purple line in Fig.

Validation of the convex hull from the OQMD

2(b)] for

is discussed in the Supporting Information.

the chemical decomposition tendency serves to

The formation energies of Ybx Co4 Sb12 , with respect to the

x = 0

and

x = 1

2(b).

x = 1/4

is

≈ −44

meV. Therefore,

limit the solubility of Yb in CoSb3 , as has been

endmembers,

suggested by previous theoretical works.

computed via cluster expansion, are shown in Fig.

For example, the formation energy for

6669

In experiments, a phase of Ybx Co4 Sb12 with

The cluster expansion, which is

a maximum in Yb content is typically achieved

t to 40 structures, achieves a leave-one-out

without evidence of Yb ordering or separation

cross-validation score of 1.6 meV per lattice

into Yb-rich and Yb-poor phases; this phase of-

site.

ten coexists with the binary impurity phases

Additional details on the cluster expan-

42,44,7073

sion are contained in the Supporting Informa-

CoSb2 and YbSb2 .

tion. A mild phase separating tendency is ob-

qualitatively consistent with our computational

served, with positive formation energies on the

ndings. We note that samples whose prepara-

order of tens of meV per lattice site. Phase sep-

tion involves ball milling may exhibit a inhomo-

aration has also been predicted in Lax Fe4 Sb12

geneous Yb distribution due to non-equilibrium

via coherent potential approximation calcula-

eects, however.

tions (in this case with appreciable energy of mixing

∼

0.6 eV),

37

This behavior is

74

Although it has received considerable atten-

whereas a previous cluster

tion in the literature, the solubility (FFL) of Yb

expansion for Bax Co4 Sb12 found several stable

in CoSb3 remains controversial, with values re-

ordered phases (formation energy no lower than

ported ranging from 0.2 to 0.7.

∼ −90

Among the ordered phases,

we address this important issue by computing

we nd (110) and (100) superlattices are the

the solubility from our rst-principles calcula-

lowest-energy structures (still higher in energy

tions and comparing directly to past theoret-

than phase separation).

ical calculations and recent experiments.

meV).

65

4143,66,73,75

Here,

In

The SQS exhibit DFT-computed formation

our work, we take a subregular solid solution

energies (615 meV) similar to the formation

model (corresponding to the polynomial t to

energies of the ordered structures, which is re-

SQS energetics discussed above) and quantita-

ective of the relatively weak interaction be-

tively incorporate ideal congurational entropy;

tween the rattlers.

vibrational entropy is discussed below. To com-

Therefore, given the small

magnitude of the formation energy,

we ex-

pute the solubility, we employ the common tan-

pect congurational entropy to easily overcome

gent construction with respect to the binary de-

the phase separation tendency at reasonable

composition products observed in experiment

synthesis temperatures and enable single-phase

(CoSb2 and YbSb2 ).

solid solutions of Ybx Co4 Sb12 .

For example,

evaluation of the solubility from our calcula-

the ideal congurational entropy contribution

tions, as well as details on the comparison data


Further details on the


Page 6 of 17

from past theoretical studies, are included in the Supporting Information. Figure 3 illustrates our computed (thick solid black

line)

temperature-dependent

solubility

curve (solvus) for Yb in CoSb3 in comparison

Theory: This work

with past theory and experiments.

Theory: This work with ∆S=1.9kB scaling

solid orange line) and 69 (thin dashed purple

Theory: Shi 2005

line) found relatively small temperature depen-

Theory: Shi 2008

dence.

In addition, the similar previous the-

oretical work of Mei et al.

Experiment: Co−rich data

reported a sin-

gle FFL value (0.30) rather than temperature-

Experiment: Sb−rich data

dependent results.

67

The lack of strong temper-

ature dependence in the past theoretical works

1300

is in disagreement with the experimental results

1200

of Tang et al., who observed the measured solu-

1100

bility can vary by as much as a factor of ve as

1000

T (K)

The pre-

vious theoretical calculations of Refs. 66 (thin

a function of annealing temperature.

900

42

Impor-

tantly, the past theory works also found nite

T → 0. T → 0 (third

solubility for

800

ish for

700

Since entropy must vanlaw of thermodynamics),

such a solid solution cannot exist on the

T =0

600

phase diagram.

100

ical predictions violate the third law. They also

76

In this sense, the past theoret-

are inconsistent with experiments, which sug-

0 0

0.1

0.2

0.3

0.4

gest FFL approaching 0 for

0.5

T → 0. 42

We note

that it may be possible to maintain a nite rat-

x in YbxCo4Sb12

tler concentration at low

T

in experiment, but

only as a result of kinetic eects. In contrast to the past theoretical works, we

Figure

3:

Temperature-dependent solubility

correctly nd a vanishing solubility at low tem-

(lling fraction limit) of Ybx Co4 Sb12 . The com-

perature, which is consistent with the third

puted solvus from this work is shown without

law and agrees with experiment.

(thick solid black line) and with (dashed grey ∆S/kB line) the scaling factor e for a vibrational

shows a quantitative comparison of our com-

∆S = 1.9 kB ,

as discussed in

42 for the solubility in the Co-rich (red open

the main text. Past theoretical solubility pre-

triangles) and Sb-rich (blue open squares) re-

dictions from Refs. 66 (thin solid orange line,

gions of the experimental phase diagram.

labeled Shi 2005) and 69 (thin dashed purple

focus on the Co-rich data since it corresponds

line, labeled Shi 2008) are included for compar-

to equilibrium of Ybx Co4 Sb12 with CoSb2 and

ison.

YbSb2 .

entropy change

42

Figure 3

puted solvus with experimental data from Ref.

Experimental data for the Co-rich (red

42

We

Our computed values appreciably un-

open triangles) and Sb-rich (blue open squares)

derestimate the experimental solubility.

region of the experimental phase diagram are

result, although we nd a larger temperature

taken from Fig. 4 of Ref. 42.

dependence compared to past theoretical works

As a

(and the correct exponential dependence in the dilute limit), our computed temperature dependence is still signicantly smaller than experiment.

42

This solubility underestimation has

been commonly observed in rst-principles pre-


Page 7 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Chemistry of Materials The electronic band structure of the endmem-

dictions due to the neglect of vibrational en-

7780

tropy.

bers (x

Given Yb is a rattler, corresponding

= 0

x = 1)

and

is shown in Fig.

to low-frequency vibrational modes, one can ex-

Filling the voids with Yb (x

pect the inclusion of vibrational entropy to sig-

metallic state with carriers in the conduction

nicantly stabilize the Ybx Co4 Sb12 solid solu-

band.

tion, enhancing the solubility.

to gap midpoint at

Via tting, we

nd that a vibrational formation entropy

∆S

and

> 0)

4.

leads to a

For comparison, the zero of energy set

Γ. Here, both the x = 0 x = 1 structures are fully relaxed, so x = 1

(of the solid solution with respect to the linear

has a smaller Brillouin zone volume than that

combination of binary decomposition products)

of

of a

1.9 kB , taken to modify the solubility e∆S/kB multiplicative factor (dilute limit

x=1

due to a larger lattice parameter. We

via

note that we nd the same trends discussed be-

ap-

low if we x to the relaxed

x = 0 lattice param-

proximation), is sucient to reconcile the the-

eter.

oretical prediction with experiment, as shown in Fig. 3. Experimental measurements and/or

− With Co contributing 9 e and Sb3 contribut− − 3 ing 9 e , CoSb3 satises the 18 e rule and

calculations of the phonon entropy will be im-

forms a semiconductor with a small experimen-

portant future work towards achieving quanti-

tal band gap on the order of 3550 meV.

tative solubility prediction for skutterudite ma-

We nd CoSb3 is a direct-gap semiconductor,

terials.

81,82

We note that, beyond vibrational en-

with the singly-degenerate valence band and

tropy, non-ideal congurational entropy may

triply-degenerate conduction bands located at

also help reconcile the computational results

Γ,

with experiment.

The valence bands are primarily a mix of Co

consistent with previous calculations.

p/d

Endmember electronic band structure

p

and Sb

5,83,84

character, whereas the conduc-

tion bands are primarily Co

d

character.

As

shown in early electronic structure calculations on CoSb3 ,

83,84

the valence band and one of the

conduction bands exhibit linear dispersion (as

1

opposed to the usual parabolic behavior) near the band extrema. Filling the voids of CoSb3 with Yb leads to

0.5

two major eects.

First of all, it can be seen

that adding Yb increases the magnitude of the band gap.

Energy

0

(eV)

the emergence of additional conduction band

CoSb3

minima close in energy to the band edge at

Yb1/4CoSb3

Γ.

We observe such bands, which we refer to as emergent bands for reasons discussed in the

−0.5

next section, at four locations along the highsymmetry

−1

Secondly, the Yb rattlers lead to

Γ H

Γ

N

Γ

P

H

k -path

shown in Fig. 4: (1) between

and H, (2) at N, (3) between P and

Γ,

and

(4) between P and H. We note that another emergent band minima exists between N and

Figure

4:

Electronic

band

structure

H, not shown in Fig. 4. For suciently large

of

Ybx Co4 Sb12 for the fully-relaxed endmembers (x

=0

and

x = 1).

energy for arrow.

x=1

x=1

responds to the band at

Energies are plotted with

respect to the gap midpoint at

x,

the conduction band minimum no longer corat

Γ, and the Fermi

Γ

Γ,

i.e., the direct gap

is no longer the smallest gap.

The same

trends of band gap opening and new, emergent

is indicated by the horizontal

conduction band minima are also present for

Emergent conduction band minima in

intermediate

are noted by the vertical arrows.

x

values, as discussed below.

Here, we discuss the band gap trends in more



Page 8 of 17

Γ valence and conduction bands the k -space directions along the

detail. Although semilocal DFT is well known

is found for the

to exhibit errors in band gaps,

along each of

85

quasiparticle

and spin-orbit coupling corrections have been

computed high-symmetry path in the Brillouin

shown to yield only small changes to the gap

zone, as is discussed in the Supporting Informa∗ tion. The decrease in carrier mobility (∼ 1/m )

(∼

0.1

eV) in this system,

86,87

and we expect

the computed trends to be valid even if there are

for larger

x

is consistent with experiments.

74

small errors in the absolute values. The com-

Fully

Electronic structure of partiallylled skutterudite CoSb3

lling the voids with Yb (corresponding to the

In order to probe the electronic properties of

x=1

Ybx Co4 Sb12 with intermediate

puted band gap of CoSb3 , 0.155 eV, is larger but still comparable to the small experimental band gap on the order of 3550 meV.

81,82

structure) increases the gap to 0.210 eV.

x (0 < x < 1),

The band gap increases further to 0.239 eV if

we compute the eective band structures for

we x to CoSb3 structural parameters; this in-

structures with partial Yb lling, as shown in

dicates the gap opening is a chemical, not struc-

Fig. 5. For ordered structures, we choose the

tural, eect. In order to understand the role of

low-energy structures corresponding to (110)

the Yb atoms, we also articially dope CoSb3

superlattices since such structures have a rela-

by increasing the electron chemical potential of

tively small primitive unit cell. In addition, we

CoSb3 and adding compensating homogeneous

note that previous DFT calculations found that

background charge to retain charge neutrality,

(110) is the lowest-energy surface.

rather than including Yb atoms. In such arti-

pare to disordered structures in order to assess

cially doped CoSb3 , we nd a substantial band

the eect of Yb ordering on the electronic struc-

gap of 0.315 eV, which indicates that gap open-

ture.

ing upon doping is not specically tied to the

tures show similar eects as the fully-lled skut-

presence of Yb as the rattler. Similar behavior

terudite material: band gap opening and emer-

was found in a previous study of Ba-lled skut-

gent conduction band minima.

terudite CoSb3 ,

which suggests the eect is

rattler ordering does not have a dramatic eect

largely invariant to the nature of the rattler. In

on the band structure, though the presence of

contrast to the band gap opening behavior, the

the rattler atoms is necessary to achieve the new

emergence of new, low-energy conduction band

conduction band minima (as discussed above).

minima is not found for the articially-doped

We note that there are dierences between the

x = 1

ordered and disordered structures in the ner

88

case, whose band structure is shown

in the Supporting Information. This indicates

89

We com-

Both the ordered and disordered struc-

This suggests

details of the emergent bands.

that the presence of the rattler atoms is respon-

In order to investigate the nature of the elec-

sible for the emergent conduction band minima.

tronic states in the partially-lled skutteru-

Filling the voids with Yb also impacts the ∗ carrier eective masses m determined via a

dite material, we compute for the superlat-

quadratic t of the band structure near the

tions (1) on all atoms corresponding to the Yb-

band extrema.

rich region (which we call

tice structures the projections of the wavefunc-

The eective masses become

larger (corresponding to less dispersive bands)

and (2) on

all atoms corresponding to the Yb-poor region

x = 0. For example, for the valence band at Γ along the P direction, the eective mass is 0.06 me for x = 0 as compared to 0.09 me for x = 1. Along this di-

(which we call

rection, there are two heavy and one light con-

clude those lying in the purple planes drawn in

x = 0, the corresponding eective masses are 0.19 me and 0.07 me , appreciably smaller than the 0.21 me and 0.11 me for x = 1, respectively. The same qualitative trend

Fig.

for

x=1

pYb−rich )

as compared to

duction band.

pYb−poor ).

The layers of atoms

at the interfaces between these regions are not included in either of these projections. For example, for

For

x = 1/2,

these interface atoms in-

5(d) as well as the corresponding atoms

between layers 2 and 3. We dene the spatial polarization, with respect to the Yb-rich and


Page 9 of 17 0.4 0.2

(a) x=1/4

0

x=0 x=1

Disordered Superlattice

−0.2

0.4 0.2

0.4 Energy (eV)

0.2

(b) x=1/2

0 −0.2 0.4

x=0

−0.2

Superlattice

x=1

(a) x=1/4 Yb−rich

0.2

Spatial Polarization

0 (b) x=1/2

−0.2

0.2

(c) x=3/4

0

Yb−poor

0.4 0.2

−0.2 H

Γ

[110]

0

0.4 Energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


N

Γ

P

0

H

(d) Superlattice structures layer 1

[110]

layer 2

layer 3

(c) x=3/4

−0.2 H

Γ

layer 4

N

P

Γ

H

Figure 6: Electronic band structure of the ordered (110) superlattice of Ybx Co4 Sb12 for (a)

x = 1/4,

(b)

x = 1/2,

and (c)

x = 3/4.

Size

of the points is proportional to the weight in the eective band structure as determined by band unfolding, and the color indicates the spatial polarization

ξ

of the wavefunctions (as de-

ned in the main text).

We only plot points

with weight greater than 0.02 (for visual clarity)

pYb−rich

whose

dered (110) superlattice and disordered struc-

than 0.025 (to avoid numerical errors in com-

ture of Ybx Co4 Sb12 for (a)

x = 1/4,

(b)

puting

x =

ξ ).

and

pYb−poor

Figure 5: Electronic band structure of the or-

are both greater

Fermi energies are indicated by hori-

Size of the points is

zontal arrows. The endmember band structures

proportional to the weight in the eective band

are shown as solid lines. All structures are xed

structure as determined by band unfolding. For

to the relaxed

1/2,

and (c)

x = 3/4.

visual clarity, we only plot points with weight greater than 0.02. Fermi energies are indicated by horizontal arrows.

The endmember band

structures are shown as solid lines. tures are xed to the relaxed parameter.

All struc-

x = 0

lattice

The superlattice structures corre-

spond to retaining Yb in layer 1 forACS x = 1/4, Plus Environment Paragon layers 1 and 2 for 3 for

x = 3/4

x = 1/2,

and layers 1, 2, and

as shown in panel (d).

9

x=0

lattice parameter.


Yb-poor regions, of the electronic states as

minimum for

Γ

pYb−rich /x . ξ= pYb−rich /x + pYb−poor /(1 − x) The factors of

1/x

1/(1 − x)

and

x 6= 1/2.

x = 0

the literature for skutterudites

and N for

strongly suggest that several other band minima, including those emergent bands absent in the unlled material, should also substantially

This

contribute to the electronic transport. Indeed,

Γ

as shown in Fig. 5(a), even for the lowest lling value considered of

one can think of Ybx Co4 Sb12 as containing two

in the literature.

regions from the emergent bands. use

of

our

the

term

results

Conclusions

in

the context of the concept of band convergence. from

The behavior we nd is quite distinct typical

PbTe1−x Sex ,

90

band

convergence

Mg2 Si1−x Snx ,

and Pb1−x Srx Se

93

91

(such

these other

than the band minimum previously considered

Γ and electrons more localized in the Yb-rich

put

x = 1/4,

band minima are signicantly lower in energy

distinct types of carriers: delocalized electrons

our

Γ to N in Fig. 4) to Γ, this

vergence term, as discussed above. Our results

are much more spatially delocalized. Therefore,

and

(i.e., the lowest-energy mini-

corresponds to the typical use of the band con-

to the emergent bands, the electronic states at

discuss

Γ

that of the conduction band minima at

as cations and donate their charge. In contrast

bands

x=0

mum roughly halfway from

Although

are not localized on the Yb atoms, which act

we

takes a dif-

existing conduction band minimum between

signicantly larger

in the Yb-rich region, we note that the states

Finally,

5,6,94

plied to the convergence of the energy of an

these states have a strong preference to localize

emergent

4.

Since in these previous works the term was ap-

than 1/2, which indicates such states tend to

at

Γ)

eV in Fig.

ferent form than that identied in our work.

tice band structure. The emergent conduction

indicates that the highly-dispersive bands at

∼ 0.5

The band convergence previously discussed in

6, which shows the superlat-

much closer to 1/2.

band at energy of

than convergent bands.

electronic states is shown via the color of the

ξ

In other

new band minima as emergent bands rather

The spatial polarization of the superlattice

show values of

x = 0.

converge with Yb lling and we describe the

value

region.

Γ

5(a-c), es-

In this sense, such bands emerge rather than

ization on an atom in the Yb-rich (Yb-poor)

be localized in the Yb-rich region.

in Figs.

originate, which is the very at (away from

of 1 (0) indicates a 100% preference for local-

ξ

x

similar to the band from which it appears to

A

localization on an atom in the Yb-rich region as

bands exhibit values of

for fractional

words, this band minima does not appear at all

wavefunction exhibits an equal preference for

points in Fig.

k -space

sentially does not exist for

spatial polarization value of 1/2 indicates the

ξ

of the way from

to P), which is associated with the lowest-

in

are included

that in the Yb-poor region, whereas a

x = 1 (∼ 2/3

energy minima along this high-symmetry line

to normalize for the diering sizes of the Ybrich and Yb-poor regions when

Page 10 of 17

as

Pb1−x Mgx Te,

Using rst-principles calculations, we provide

in

a detailed understanding of the phase stability

92

and electronic properties of lled skutterudite

) in which the energies of

CoSb3 . The Yb-lled skutterudite Ybx Co4 Sb12

multiple existing band minima converge as a

exhibits a mild tendency to phase separate into

function of some tuning parameter, such as

the Yb-rich and Yb-poor endmembers, as well

temperature or doping. In our case, several of

as a strong tendency for chemical decomposi-

the low-energy minima away from

Γ

are not in

tion into CoSb and YbSb binaries.

Single-

general even present (in any recognizable form)

phase solid solutions with a limited Yb solubil-

in the unlled material.

ity, observed in experiment, are stabilized by

Γ

For example, as can be seen in Fig. 4 between

congurational entropy. In addition to enhanc-

and P, the lowest-energy conduction band

ing the band gap and eective masses, the pres-


Page 11 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ence of Yb leads to two distinct types of electronic carriers:

(3) Langmuir, I. Types of Valence. Science

1921, 54, 5967.

(1) new emergent conduction

band minima whose electronic states are local-

(4) Snyder,

ized near the rattler atoms and (2) the delo-

J.;

Toberer,

E.

S.

Com-

plex thermoelectric materials. Nat. Mater.

calized electronic states at the Brillouin zone

2008, 7, 105114.

center.

Acknowledgement

G.

We thank Je Snyder

(5) Tang, Y.; Gibbs, Z. M.; Agapito, L. A.;

(Northwestern) and Wenjie Li (Penn State) for

Li, G.; Kim, H.-S.; Nardelli, M. B.; Cur-

useful discussions.

We acknowledge support

tarolo, S.; Snyder, G. J. Convergence of

from the U.S. Department of Energy under

multi-valley bands as the electronic ori-

Contract DE-SC0014520.

gin of high thermoelectric performance in

Computational re-

sources were provided by the National Energy

CoSb3 skutterudites. Nat. Mater.

Research Scientic Computing Center (U.S.

14, 12231228.

Department

of

Energy

Contract

DE-AC02-

2015,

(6) Hanus, R.; Guo, X.; Tang, Y.; Li, G.;

05CH11231), the Extreme Science and Engi-

Snyder, G. J.; Zeier, W. G. A Chemical

neering Discovery Environment (National Sci-

Understanding of the Band Convergence

ence Foundation Contract ACI-1548562), and

in Thermoelectric CoSb3

the Quest high performance computing facility

Skutterudites:

Inuence of Electron Population, Local

at Northwestern University.

Thermal Expansion, and Bonding Interactions. Chem. Mater.

Supporting Information Avail-

2017, 29, 11561164.

(7) Rogl, G.; Rogl, P. Skutterudites, a most

able

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Electronic Structure and Phase Stability of Yb-filled CoSb3

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