Energetic Stability and Thermoelectric Property of Alkali-Metal-EncapsulatedType-I Silicon-Clathrate from First-Principles Calculation

Kaoru Nakamura+, Susumu Yamada and Toshiharu Ohnuma

Central Research Institute of Electric Power Industry, Yokosuka 240-0196, Japan

Possible combinations of alkali metal guest atoms and substitutional group-13 atoms in type-I Si clathrate and their thermoelectricproperties were investigated using first-principles calculations. All alkali metals could be encapsulated as the guest element into a Si46 cage, andeither Al or Ga was suitable for a substitutional atom. From the formation energy, possible clathrate compositions were selected as K8Al8Si38,K8Ga8Si38, Rb8Al8Si38, Rb8Ga8Si38, Cs8Al8Si38 and Cs8Ga8Si38. The thermoelectric properties of these compositions were calculated asfunctions of temperature and carrier density, using the Boltzmann transport equation and the calculated band energy. The obtained dependencesof the Seebeck coefficient and electrical conductivity on the carrier density were discussed from the viewpoint of band structure. Thethermoelectric properties were optimized to maximize ZT for each composition by controlling the carrier density. ZT µ 0.75 was predicted as thehighest ZT value for hole-doped Cs8Ga8Si38. [doi:10.2320/matertrans.MBW201204]

(Received November 8, 2012; Accepted December 7, 2012; Published January 25, 2013)

Keywords: thermoelectric, first-principles calculation, clathrate

1. Introduction

In recent years, reusable and clean energy generationtechnologies have been attracting much attention. Thermo-electric conversion is one of the promising clean energysources because it enables the conversion of wasted heat intoelectric power.

In 1994, Slack proposed the concept of Phonon GlassElectron Crystal (PGEC).1) This hypothetical material cansimultaneously behave as glass for phonons and as crystal forcarriers. Filled skutterudite and clathrate compounds are niceexamples of Slack’s PGEC concept. Such compounds arecomposed of a characteristic atomic structure. “Host atoms”form a cage network, and good electrical conductivity isexhibited because of the covalent bonding of the cagenetwork. Cages are usually formed to encapsulate theadditional element, which is frequently referred to as a“guest atom”. The guest atom is weakly bound in the cageand vibrates like an Einstein oscillator. The characteristicvibration of the guest atom can scatter the lattice vibration ofthe cage. This concept is referred to as “rattling”. Therefore,cage-networked materials such as glassy materials generallyexhibit quite low thermal conductivities.1­9)

Thermoelectric materials can generate electric powerthrough the Seebeck effect. Its conversion efficiency isdetermined from the dimensionless figure of merit (ZT),defined as follows:

ZT ¼ s2·T=ðke þ klÞ; ð1Þwhere s is the Seebeck coefficient, · is electrical conductivityand T is temperature. ke and kl are electrical and latticethermal conductivities, respectively. Generally, ZT > 1 isregarded as a standard for actual application of thermoelectricconversion. It is crucial to increase both the Seebeckcoefficient and the electrical conductivity while suppressingthe thermal conductivity in order to achieve high thermo-electric performance. PGEC materials exhibit both highelectrical and low thermal conductivities and have been

extensively investigated for application as thermoelectricmaterials. In order to achieve high Seebeck coefficient, a lowcarrier density is desirable because the Seebeck coefficient isdetermined from the transport of entropy per carrier. On thecontrary to the Seebeck coefficient, high electrical conduc-tivity can be achieved with high carrier density. Thus, thereis a tradeoff relation between the Seebeck coefficient andelectric conductivity, and the optimal carrier density must befound to maximize ZT.10)

Carrier density is frequently optimized in semiconductormaterials by adding a dopant. Clathrate compounds arethought to be one of the intermetallic compounds that exhibita Zintl phase11) in which a guest atom supplies an electronwhile the substitution of an aliovalent element into a cage sitecan compensate for the supplied electron. If the number ofguest and substitutional atoms is properly controlled, chargeneutrality of the system is preserved. The carrier density ofsuch an intrinsic semiconductor can thus be controlledthrough adjusting the composition of guest and substitutionalatoms. Among the clathrate compounds, Ge-based clathratehas been widely investigated5­9) because its localized d-electrons give it favorable thermoelectric properties. Si-basedclathrate, on the other hand, has not been investigated ascompared with the Ge-based one for use in thermoelectricapplications.

For example, reports on the ZT of Si clathrate are quitelimited. Kuznetsov et al. estimated the ZT of Ba8Ga16Si30 tobe 0.87 at 870K,13) and C. Candofi et al. reported that a ZT ofapproximately 0.2 had been achieved at 600K for Ba8Aux-Si46¹x, where x = 5.59.14) It has not been established whetherthe maximum ZT of Si-based clathrate exceeds 1. Si is aubiquitous element that is cheaper than Ge and is nontoxic.Therefore, we believe that if a stable Si-based clathratewithout harmful and rare-earth element will be obtained, itsrange of applications is significant, even if it only exhibiteda moderate ZT. Experimental optimization of additionalelements and the carrier density would require an enormousnumber of research trials, in general. On the other hand, anumerical method of predicting the thermoelectric propertiesof materials on the basis of theoretical calculations has been+Corresponding author, E-mail: n-kaoru@criepi.denken.or.jp

Materials Transactions, Vol. 54, No. 3 (2013) pp. 276 to 285©2013 The Japan Institute of Metals

developed. If such a simulation is used to optimize thecomposition of the compound and estimate the maximum ZT,is would help to develop effective thermoelectric materials.

In this study, several possible combinations of guest andsubstitutional atoms for the Type-I Si clathrate and theirthermoelectric properties were systematically investigatedusing first-principles calculation. Alkali metals were consid-ered as guest elements because thermoelectric properties inalkali metal encapsulated Si clathrates have not been revealedalthough their electronic structures were investigated.15)

Reducing the number of substitutional elements is expectedto reduce the amount of alloy scattering of the carriers morethan using alkali earth elements for the guest elements would.On the basis of the calculated thermoelectric properties, wewill discuss the optimal composition of guest and substitu-tional atoms that maximizes ZT and whether ZT > 1 can beachieved for Si clathrate.

2. Computational Procedure

A plane-wave-basis projector augmented-wave (PAW)method implemented in Vienna ab initio simulation package(VASP) code16) was used for structural relaxation and precisedetermination of the total energy. The exchange-correlationfunctional was defined using a generalized gradient approx-imation (GGA) of the Perdew­Burke­Ernzerhof (PBE)form.17,18) The plane-wave cutoff energy was set to 300 eVthroughout the calculation. The Monkhorst­Pack scheme19)

was used for k-point sampling. The numerical error wasestimated to be 1meV/atom for a Si46 unit cell on the basisof convergence tests of the cutoff energy and k-point mesh.A force of <10¹2 eV/¡ and a total change in energy of10¹8 eV were selected as the convergence criteria for thestructural optimization.

The formation energy (Ef ) for guest atom encapsulationand host atom substitution was calculated using the followingequation:

Ef ¼ ETðdopant: qÞ � fETðperfectÞ þ�nA®A ��nB®Bgþ qð¾F þ EVBMÞ; ð2Þ

where ET(dopant: q) and ET(perfect) are the total energiesobtained from the first-principles calculations for the dopantand the perfect cell. q is the charge state of the dopant. Inthe whole cell, charge balance was maintained by adding/removing electrons. In eq. (2), nA and ®A are the number andchemical potential of added dopant A, respectively. If thedopant is considered as a substitutional atom, nB and ®B mustbe considered because they represent the number of removedhost Si atoms and the chemical potential of Si in accordancewith dopant substitution. ¾F is the Fermi energy, whichcorresponds to chemical potential of the electron. EVBM is theenergy of the valence band maximum. Here, EVBM wasdetermined using the following equations:

EVBM ¼ EVBMðperfectÞ þ VavðdefectÞ � VavðperfectÞ ð3Þand

EVBMðperfectÞ ¼ ETðperfect:0Þ � ETðperfect:þ1Þ; ð4Þwhere Vav is the average electrostatic potential. When adopant is introduced, Vav(defect) is defined as the average

electrostatic potential at the farthest site from the dopant. Thistreatment can correct distortion of the band structure inducedby long-range interaction of an ionized dopant. ET(perfect:0) andET(perfect:+1) are the total energy of the perfect cell when thecharge is neutralized and when one electron is removed,respectively. The Fermi energy (¾F) in eq. (2) was measuredfrom the EVBM determined in eq. (3), and it can vary withinthe band gap. Chemical potentials ®i of element i werecalculated as per atom energy in each primitive cell. Si46 unitwas used for the perfect cell throughout the calculation. Itwas confirmed that the difference in the formation energybetween the unit cell and the 2 © 2 © 2 expanded supercellwas about 0.1 eV. This difference does not affect our resultsand conclusion.

After the structural relaxation, the detailed band structurewas further calculated using the all-electron method based onthe Full potential Linearized Augmented Plane Wave +Local Orbital (FLAPW+lo) approach implemented in theWIEN2k code.20) Rkmax, determines the size of the basis set,was set to 7, which assures convergence of total energy toless than 0.01meV/atom for Si46. A 12 © 12 © 12 k-pointmesh was used to sample the band energy within theBrillouin zone. The exchange-correlation functional wasapproximated using a GGA of Engel­Vosko formalism (EV-GGA),21) which is known to reproduce the bandgap of Si andis previously used for the first-principles calculation of theSeebeck coefficient of the Zn-substituted Ge-clathrate.22)

On the basis of the band energies obtained at each k-point,the band energy was interpolated using a 6-times denser gridthan the original k-point mesh. The thermoelectric properties,including the Seebeck coefficient, electrical conductivity,and electronic thermal conductivity were calculated bynumerically solving the Boltzmann transport equation usingthe BoltzTraP code.23) A constant carrier relaxation timeapproximation was assumed so that the Seebeck coefficientcould be obtained independent of the carrier relaxation time,while the electrical conductivity and electronic thermalconductivity could be obtained as carrier-relaxation-time-dependent quantities. Carrier doping was treated as a shift inthe Fermi level on the basis of the band structure of theX8Y8Si38. Such a treatment is called a “rigid-band approx-imation”. However, as the band structure is expected toseverely change in heavy doping, the rigid-band approxima-tion would fail when Fermi-level is located more than 0.1 eVinto the band edge.24) We also checked the applicability ofrigid-band approximation to the present Si-clathrate system,from the analysis on band structures of one electron and/orhole doped Si46 and K8Ga8Si38. It was confirmed that changesof band structure could be characterized only by shifting theFermi level, which was moved about 0.1 eV inside bands.Therefore, the shift of the Fermi level was limited betweenthe energy level of 0.1 eV inside each band edge. Thermalactivation of the carrier was considered using the Fermi­Dirac distribution function.

Our research group had already investigated the latticethermal conductivity of the Si clathrate with moleculardynamics simulation and the Green­Kubo method.25­27) At300, 600 and 900K, a 4 © 4 © 4 expanded clathrate supercellwas equilibrated and then 12-ns simulation was performedwithin the NVE ensemble, and thermal flow was integrated.

Energetic Stability and Thermoelectric Property of Alkali-Metal-Encapsulated Type-I Silicon-Clathrate from First-Principles Calculation 277

The interatomic potential was constructed to reproduce thepotential energy curve for the guest atom within a Si cage.(See details in Refs. 26) and 27).) The calculated latticethermal conductivity was merged with the present first-principles-based results, and ZT of each Si clathrate was thenpredicted.

3. Results and Discussion

3.1 Formation energy3.1.1 Alkali metal and group 13 element in Si46

A schematic illustration of the type-I Si clathrate, Si46, isshown in Fig. 1. The atomic structure and charge density plotshown later were drawn using VESTA software.28) Si in theclathrate structure has the same 4-fold coordination similar tothat in diamond. The primitive cell is cubic and is composedof two Si dodecahedra (12-Si-atom cage) and six Sitetrakaidecahedra (14-Si-atom cage). The Wyckoff represen-tations of independent atomic positions of the cage site (6c,16i and 24k) and the internal cage site occupied by the guestelement (2a and 6d) are also shown in Fig. 1. The 2a and 6dsites are center of the 12- and 14-Si-atom cages, respectively.The calculated lattice constant of Si46 was 10.225¡, similarto the previously obtained theoretical value, 10.350¡29). Thedifference in energy between the Si46 and diamond Si was

63meV/atom, which is very consistent with the previouslyobtained theoretical value of 69meV/atom.29) The tinyenergetic difference is thought to originate in the strain ofSi­Si bonds.

In the concept of the Zintl phase, one alkali metal suppliesone electron when it is encapsulated in a Si cage. However,one group-13 element atom substituted at a Si-cage sitecompensates for the one electron because it has one fewervalence electron than Si. Therefore, the alkali metal X in Si46is thought to be X+ in Si46, while the substitutional group-13element Y is thought to be Y¹ in Si46. The calculatedformation energies of alkali metals and group-13 elements areshown in Figs. 2(a) and 2(b), respectively. As the formationenergy of the charged dopant is dependent on the chemicalpotential of the electrons, we assumed that the Fermi energywas located at the center of the band gap. This situationcorresponds to the intrinsic semiconductor. All alkali metaldopants exhibit lower formation energies for the guestelement than for substitution. This order did not changeeven though the Fermi level was located at the top of thevalence band (p-type doping limit) or at the bottom ofthe conduction band (n-type doping limit). The formationenergies of K, Rb and Cs at the 2a site gradually increasedwith increasing ionic radius. In the pristine Si46, thetheoretical size of the 12-Si cage is 3.32¡, while that of14-Si cage is 3.68¡. Therefore, larger elements such as K,Rb and Cs are more comfortable at the 6d site than they are at2a site. For group-13 elements, it seems that the interstitialformation energies at 2a and 6d site decrease as going downthe periodic table. Especially for Al and Ga, the differencebetween interstitial (2a and 6d site) and substitutional (6c, 16iand 24k site) formation energies is fairly small. Figures 3(a)and 3(b) show the Fermi-level dependence of the Al andGa formation energy in Si46. The gradient of each linecorresponds to the charge state, and the transition levels arerepresented as symbols. The previously mentioned formationenergies in the discussion for Figs. 2(a) and 2(b) werecalculated on the assumption that the Fermi level was locatedat the middle of the band gap. However, the Fermi level canvary within the band gap in accordance with dopant addition.Let us consider the situation in which the guest element isfirst encapsulated in Si46 and then Al or Ga substitutionoccurs. As the clathrate structure is thought to be formedas Si encloses the guest element, the Fermi level should belocated near the bottom of the conduction band. Obviouslythe substitutional formation energy is lower than the

Fig. 1 Crystal structure of type-I clathrate Si46. Crystallographic sites arerepresented with Wyckoff notation.

(a) (b)

Fig. 2 Formation energies of (a) alkali metals and (b) group-13 elements at each crystallographic site in Si46.

K. Nakamura, S. Yamada and T. Ohnuma278

interstitial formation energy for both Al and Ga, and the fullyionized (Al+Si and Ga+Si) states are stable in n-doping.Therefore, when these elements are doped with alkali-metalelements, the group-13 element can be substituted into theSi-cage site.3.1.2 Comparison with literature data on the synthesis

Except for Li, all alkali-metal-encapsulated Si clathratesexperimentally synthesized exhibit the type-I clathratestructure: Na8Si46,30) K7.62Si46,31) Rb6.15Si4631) andCs7.6Si46.32) For Al and Ga, no report on the synthesis ofsingle-doped clathrate is available, but there are reports of thesubstitutional element co-doped with guest element, such asBa8Al16Si3033) and Ba8Ga16Si30.13) For Li and Tl, there havebeen no report on experimental synthesis of Si-clathrate.Although Li exhibited a lower interstitial formation energythan substitutional formation energy, the phase diagram ofLi­Si shows that phase separation of LiSi and Si occurs at470K.34) It has been reported for In that Ba8In16Ge3035) andK8In18Si2836) were synthesized. Although Tl is very toxicelement, its calculated interstitial formation energy is lowestamong all group-13 elements. The electronegativity of Tl is2.04 (based on Pauling criterion), which is two-times largerthan that of Si. Encapsulated Tl in the Si-cage is thought to bestably bonded with Si because its ionic radius is two-timeslarger than that of Si. Therefore, low thermal conductivity isnot expected for encapsulated Tl because rattling effect arisesfrom the loosely bound guest atom with the cage.

Present results for the formation energy indicate that cagesize determines which guest elements are favorable. Forexample, Rb-doped Si clathrate has been synthesized asRb6.15Si4631) or Rb8Ga8Si38.37) The former composition isconsistent with the calculated formation energy for the 6d sitebeing more stable than that for the 2a site by 0.5 eV. On theother hand, number of the encapsulated Cs in Si46 cage was7.6,32) which is larger than that of Rb. This difference shouldbe originated in the synthesized condition, where Cs7.6Si46was synthesized under high temperature and pressure.32) Theorder of elemental stability with regard to the substitutionalformation energy can be understood by the ionic radius ofdopant. From Figs. 2(a) and 2(b), the substitutional formationenergy is the lowest at the 6c site for every element. Becausethe local strain at 6c site is largest, substitution of elementslarger than Si at the 6c site can effectively relax the elastic

strain. The substitutional formation energy of B, on the otherhand, is the lowest at the 16i site. From the chemical bondinganalysis, the 16i site exhibits the most covalent characterbecause its bond angle and length are the closest to those ofthe Si­Si bond in diamond Si. Actually, B is only substitutedat the 16i site in K7B7Si39.38)

3.1.3 Codoping of guest and substitutional atomIt frequently occurs that the donor and acceptor are

strongly bound that the formation energy is lower than that ofeach individual doped case. In Si46, various configurationsof the guest and the substitutional atom are possible. Theconfigurations were systematically examined and the lowestformation energy of the combination of one guest atom X andone substitutional atom Y were obtained. The results areshown in Fig. 4, which can be interpreted as the super-position of the interstitial and substitutional formationenergies, as shown in Figs. 2(a) and 2(b). Each guest andsubstitutional pair exhibited the lowest formation energywhen Ga was substituted at the Si-cage site. It was also foundthat in the most stable configurations all the group-13elements except for B was favorable to substitute at the 6csite. For the guest atom, Li and Na were positioned at the 2asite, while K, Rb and Cs were positioned at 6d site in themost stable guest-substitution configuration. It is noted thatformation energy of B was significantly decreased bycodoping with alkali metals because the electrons suppliedby the guest atoms contributed to the B­Si covalent bonds.

(a) (b)

Fig. 3 Formation energies of (a) Al and (b) Ga in Si46 plotted as functions of Fermi energy. Center line shows location of middle ofband gap.

Fig. 4 Formation energies of alkali metal guest atom (X) and group-13substitutional atom (Y) pairs in Si46.

Energetic Stability and Thermoelectric Property of Alkali-Metal-Encapsulated Type-I Silicon-Clathrate from First-Principles Calculation 279

The pair formation energy for Li and group-13 element, onthe other hand, was the highest. Because the ionic radius ofLi is the smallest among the alkali metals, Li moved near thesubstitutional atom during structural relaxation, which brokethe charge balance within clathrate structure. This is the mainreason that Li­Y pair is the most unstable. It must bementioned here that the other alkali metals were most stableat the center of the Si cage.

Increasing the occupancy of the guest atom encapsulatedin the clathrate cage should effectively decrease thermalconductivity. In Type-I clathrate, eight guest atoms can beencapsulated at most per chemical formula. To preservecharge neutrality, the same number of Si site as that of guestatoms must be substituted by group-13 element. For thisreason, the formation energies of X8Y8Si38 (X = Li, Na, K,Rb, and Cs, Y = B, Al, Ga, In and Tl) were calculated, andthe energetically stable compositions were analyzed forfurther band structure calculation. In the Si46 clathrate, thereare an enormous number of combinations of substitutionalpositions: more than 2 hundred billion. Previous theoreticalworks have suggested that substitutional atoms positionedat the nearest neighbor form anti-bonding, and the systembecomes energetically unstable.9,39) On the basis of thisinsight, the energies of various configurations of eightsubstitutional Ga atoms were calculated while not only theatomic positions but also the cell shape and lattice constantswere relaxed. The resulting tendencies for Ga configurationscan be separated into two groups: 1) Li and Na (small ionicradii) encapsulated in a Si cage and 2) K, Rb and Cs (largeionic radii).

Figure 5(a) shows the relative formation energy (¦Ef ) ofLi8Ga8Si38 plotted as a function of the number of Ga atomssubstituted at the 6c site. At each number of substituted Ga atthe 6c site (horizontal axis), various combinations of Ga atthe 16i and 24k site are possible. Relative energies of allexamined configurations at the fixed number of Ga at the6c site are also shown in Fig. 5. The clathrate structureapparently becomes more stable with the increasing numberof Ga atoms substituted at the 6c site because Ga substitutionmost effectively relaxes the local elastic strain at the 6c site.When the occupancy of Ga atoms at the 6c site is 1.0, Ga­Ga

bond formation can be prevented. Therefore, the most stableconfiguration within our trial was (6c, 16i, 24k) = (6, 2, 0)for Li8Ga8Si38 and Na8Ga8Si38. For the large ionic group, onthe other hand, the most stable configuration within our trialwas (6c, 16i, 24k) = (5, 0, 3) for K8Ga8Si38, Rb8Ga8Si38, andCs8Ga8Si38, as shown in Fig. 5(b). The different stable Gaconfigurations for these two groups are suggested to originatein the different sizes of the guest elements.

Figure 6 shows the local strain at the Si sites in X8Si46plotted as a function of the lattice constant. The strain at the6c, 16i and 24k sites was defined as the change in the ratiosof the average bond length in the clathrate compared to thatin diamond Si. Li is the smallest alkali metal, and the strain ateach Si site in Li8Si46 is smaller than that at each Si site inpristine Si46, resulting in a decrease of the lattice constant.The lattice constant and strain of Na8Si46 are comparablewith those of pristine Si46. As the ionic radius of the alkalimetal increases, the lattice constant and strain also increase.Because the ionic radius of Ga is larger than that of Si, Ga ismore favorable for substitution at tensile-strained Si sites.This should be the reason the substitutional formation energyis the smallest at 6c site and largest at the 16i site. In addition,

(a) (b)

Fig. 5 Relative formation energies plotted as functions of number of Ga-substituted 6c sites in (a) Li8Ga8Si38 and (b) K8Ga8Si38.

Fig. 6 Local strain of 6c, 16i and 24k sites plotted as functions of latticeconstant for pristine Si46, Li8Si46, Na8Si46, K8Si46, Rb8Si46 and Cs8Si46.Strain was determined from change in average Si­Si bond length at eachsite compared with average Si­Si bond length at each site in diamond Si.

K. Nakamura, S. Yamada and T. Ohnuma280

as the 6c site is bonded with the 24k site, further Gasubstitution at 24k sites induces unfavorable Ga­Ga bondswhen Ga is substituted at all 6c sites. Thus, the Gaconfiguration of (6c, 16i, 24k) = (6, 2, 0) was most stablefor Li8Si46 and Na8Si46 because Ga­Ga bond formation canbe prevented. However, for K8Si46, Rb8Si46, and Cs8Si46, notonly the 6c sites but also the 24k sites can effectively releasethe strain by Ga substitution. Thus, Ga substitution at all 6csites is not required for those compositions of Si clathrate.

The stable configuration of Ga found in the present study isnot perfectly consistent with the experimentally determinedGa site occupancy in K8Ga8Si38,37,40) where the Ga substitu-tional configuration was found to be (6c, 16i, 24k) = (3, 0, 5)from the analysis of Ga occupancy. Recently, Imai et al.synthesized K8Ga8Si38 single crystal,41) and occupancyof Ga at the 6c, 16i and 24k sites were 0.607, 0.026and 0.179, respectively,42) which is nearly equivalent to(6c, 16i, 24k) = (4, 0, 4). However, the present calculationis consistent with these experimental results in that Ga isdifficult to substitute at the 16i site. In addition, as shown innext section, it was found that the effect of differentconfigurations of the substitutional atom on the thermo-electric properties was negligible. Moreover, the lowestenergy configuration obtained in the present study ismore stable than the one calculated by Imai et al.15) by0.21 eV/f.u., where the previous experimentally determinedoccupancy37,40) was used.

On the basis of the above result for the stableconfigurations of substitutional atoms, the formation energiesof X8Y8Si38 per chemical formula were calculated and areshown in Fig. 7. The configurations of substitutional atomsused were (6c, 16i, 24k) = (6, 2, 0) for X = Li and Na, (6c,16i, 24k) = (5, 0, 3) for X = K, Rb and Cs, and (6c, 16i,24k) = (0, 8, 0) for Y = B because of the strongly covalentB­Si bond. The same as Fig. 4, resultant order of theenergetic stability was obtained: K > Rb > Cs > Na > Li,independent of the substitutional atom. Although the sitestrain of Rb8Si46 and Cs8Si46 was larger than that of K8Si46,as shown in Fig. 6, the large tensile strain of former twocompositions increase the energy by the stretching cage. Thisenergy increase should exceed the energy gained from thestrain released by substitutional atoms larger than Si. For thisreason, formation energies of the former two compositionsare higher than that of the K8Ga8Si38. In Fig. 7, the dottedline indicates the formation energy of K8B8Si38. Because anearly identical composition, K7B7Si39, was synthesized,38)

formation energy of K8B8Si38 can be regarded as a criterionwhether composition is formable as a clathrate or not.Therefore, X8Y8Si38 (X = K, Rb and Cs, Y = Al and Ga)was selected for further transport analysis because theirformation energies were well lower than that of K8B8Si38.

3.2 Thermoelectric properties3.2.1 Seebeck coefficient

As explained previously, optimization of the carrier densityis crucial to obtain high ZT because there is a tradeoffdependence on carrier density between the Seebeck coef-ficient and the electrical conductivity. The Seebeck coef-ficient was calculated for each composition, and the resultsfor hole- and electron-doping at 900K are plotted in

Figs. 8(a) and 8(b), respectively, as a function of the carrierdensity. The selected temperature, 900K, is almost compa-rable to the temperature of the steam temperature generatedby fossil power plants. As derived from the Wiedemann­Franz law, a Seebeck coefficient of at least ³160 µV/K isrequired in order to achieve ZT > 1. This value is representedas the dotted line in Fig. 8. Within the wide range of carrierdensities, all compositions can achieve «s« > 160 µV/K. TheSeebeck coefficient reached a maximum at the carrier densityof ³1018­1019 cm¹3. A carrier density of ³1021 cm¹3 wasneeded to maximize the Seebeck coefficient for pristineSi46. This is consistent with the change in the band gap.Introducing guest atoms into pristine Si46 decreases theband gap of pristine Si46 because the guest atom forms anadditional s-orbital energy band at the bottom of theconduction band. The lattice constant increases as the sizeof the guest atom. In accordance with this volume relaxationeffect, the degree of band dispersion is decreased, and theband gap also decreases. Actually, calculated band gapswere 0.98, 1.53, 0.76, 0.80, 0.82, 0.88, 0.91 and 0.97 eV forthe diamond Si, Si46, K8Al8Si38, K8Ga8Si38, Rb8Al8Si38,Rb8Ga8Si38, Cs8Al8Si38 and Cs8Ga8Si38, respectively, at theEV-GGA level. Although a systematic difference was foundbetween the present band gaps and those obtained by Imaiet al.15) using different functional, the present band gapsare consistent with those in the previous theoretical work.The order of carrier densities where the Seebeck coefficientreaches a maximum is consistent with the previouslymentioned order of band gaps for each composition.

It must be noted that the configurations of the substitu-tional atoms did not alter the order of the Seebeck coefficientsshown in Fig. 8. Absolute value of the Seebeck coefficientsdid not change very much. For example, the maximumSeebeck coefficients for K8Ga8Si38 with the Ga configurationof (6c, 16i, 24k) = (6, 2, 0) were 439 and ¹425 µV/Kat 900K, respectively, for the hole- and electron-dopedclathrates, while those for the configuration (6c, 16i, 24k) =(5, 0, 3) were 403 and ¹400 µV/K at 900K, respectively.In addition to this difference, the required carrier density tomaximize the Seebeck coefficient was slightly changed. Thisdifference can be explained by the different band gaps of0.80 eV for the Ga configuration of (6c, 16i, 24k) = (6, 2, 0)

Fig. 7 Formation energy of X8Y8Si38 (X = Li, Na, K, Rb or Cs, andY = B, Al, Ga, In or Tl) per chemical formula. Dotted line indicatesenergy of formation of K8B8Si38, where K7B7Si39 was reportedlysynthesized.38)

Energetic Stability and Thermoelectric Property of Alkali-Metal-Encapsulated Type-I Silicon-Clathrate from First-Principles Calculation 281

and 0.73 eV for that of (6c, 16i, 24k) = (5, 0, 3). This resultsuggests that those different configurations of the substitu-tional atom did not change the order of the resultingthermoelectric properties, and the change in the resultingZT was <0.01.3.2.2 Power factor and band structure

Figures 9(a) and 9(b) show the optimized power factor forthe hole- and electron-doped clathrates plotted as a functionof temperature. The carrier relaxation time, 2.0 © 10¹14 s,was assumed to be constant for all compositions, accordingto the estimation of Blake et al. for Ba8Ga16Si30 singlecrystal.43) Because of the balance between the Seebeckcoefficient and the electrical conductivity, the optimizedcarrier density required for maximizing the power factorwas ³5 © 1020 cm¹3, which is higher than that for theSeebeck coefficient. For the hole-doped clathrate, the powerfactor exhibits the same temperature dependence for allcompositions.

Figure 10 shows band structure of (a) K8Ga8Si38, (b)Rb8Ga8Si38, and (c) Cs8Ga8Si38. Although the configurationof the substituted Ga breaks symmetry of the pristine Si46, thesame k-point path of pristine Si46 was used for eachcomposition. The Fermi level was set at the middle of theband gap. The band structures of those compositions lookedall similar especially near the valence band edge becausethe tops of the valence bands are mainly composed ofhybridized Ga­Si, and their configuration is the same for

each composition. This should be the reason of the abovementioned temperature dependence of power factor.

The electrical conductivity exhibited a similar value whenthe hole-carrier density was higher than ³1019 cm¹3 for eachcomposition. This result indicates that the electrical con-ductivity is mainly attributed to the cage structure of Si and tothe substituted Al or Ga when the carrier is a hole. Theelectrical conductivity of the electron-doped clathrate, on theother hand, exhibited the order of Cs > Rb > K, when theelectron carrier density was higher than ³1019 cm¹3. For theelectron-doped clathrate, band dispersion near the bottom ofthe conduction band should be noted. For each composition,the bottom of the conduction band is located at the M-point.From compositions K8Ga8Si38 to Cs8Ga8Si38, the effectivemass at the M-point became lighter (Fig. 10). For the Al-substituted clathrates, such tendency did not change.Therefore, the order of the electrical conductivity wasCs > Rb > K for the electron-doped clathrates, and thereverse order was exhibited for the Seebeck coefficient at thecarrier electron density ³1020 cm¹3 (Fig. 8(b)).3.2.3 Implication for recent experimental investigation

Recently, Imai et al. synthesized K8Ga8Si38 single crystal,and showed that it exhibits an indirect band gap of 0.10 eVand semiconducting behavior.41) They suggested that thediscrepancy between the observed band gap and their calcu-lated one (0.798 eV15)) may be due to random substitutionalpositioning of Ga, vacancies, nonstoichiometry, and so on.15)

(a) (b)

Fig. 9 Temperature dependence of power factor (s2·) for (a) hole- and (b) electron-doped ternary Si clathrate in which carrier density wasoptimized to maximize power factor.

(a) (b)

Fig. 8 Seebeck coefficients for (a) hole- and (b) electron-doped ternary Si clathrate calculated at 900K and plotted as functions of carrierdensity.

K. Nakamura, S. Yamada and T. Ohnuma282

It can be considered that random Ga configuration does notsignificantly narrow the band gap because the change in theband gap was found to be at most 0.1 eV, as demonstrated bychanging the Ga configuration in the present work. Althougha number of investigated configurations was small, the bandgap should not become drastically narrower as far as thestoichiometric composition is preserved. As for the non-stoichiometry, a possible scenario is that different Gasubstitutional contents from 8. When the Ga substitutioncontent was changed from 8 to 7 or 9, the band structureswere characterized as a shift of the Fermi level inside theband, and significant changes in the band gap did not occur.This behavior was the same for K deficiency. As for the thevacancy, the calculated formation energies were 1.99 eV forthe Si vacancy at the 6c site. The lowest formation energy ofthe Ga vacancy was 2.96 eV, which is higher than that ofthe above Si vacancy. The lowest vacancy formation energyin pristine Si46 was 3.69 eV at the 6c site. The differencebetween formation energies of these Si vacancies shouldoriginate in the strain induced by the K encapsulation and theGa substitution. Therefore, Si vacancy should be dominant inK8Ga8Si38.

Band structure of K8Ga8Si37 is shown in Fig. 11(a). Thedispersion relations at the band edges are different from thosein the pristine K8Ga8Si38. In the K8Ga8Si38, there are threenearly degenerated states at the top of the valence band atthe R-point. However, because of the Si vacancy, suchdegenerated states are released, and the top of the valenceband moves to a higher energy level. Figure 11(b) showsthe contour of charge densities of the top of the valence band,cut in the plane including the Si vacancy and its nearestneighbor site. The contours are drawn within the range of0.001­0.1 e/¡3. The distance between successive contourlines corresponds to a factor of 1/10. The localized chargedensity around Si vacancy shown in Fig. 11(b) belongs to thesurrounding dangling bond. Along with the formation ofthe dangling bond, the band gap becomes narrower: about0.54 eV. Introducing further Si vacancies is expected toreduce the band gap much more. As evidence of this, thediscrepancy between the calculated and experimentally

determined band gaps can be explained by the formation ofthe Si vacancy.3.2.4 Estimation of ZT values

By merging the calculated thermoelectric properties andthe previously reported lattice thermal conductivity of X8Si46(X = K, Rb or Cs),26,27) the ZT values were predicted as afunction of temperature for all compositions. The results arerepresented as solid lines in Figs. 12(a) and 12(b), for hole-and electron-doping, respectively. The ZT values for allcompositions continuously increased with temperature. TheZT of Cs8Ga8Si38 exhibits the highest value for hole-doping,while that of Cs8Al8Si38 exhibits the highest value forelectron-doping, although the difference between the ZTvalues for Cs8Al8Si38 and Cs8Ga8Si38 is small for electron-doping.

In order to verify the present prediction, the samecalculation was conducted for Ba8Ga16Si30. The predictedZT value at 300K was 0.06 for hole-doping and 0.05 forelectron-doping. Blake et al., on the other hand, estimatedthe ZT for electron-doped Ba8Ga16Si30 at room temperature

(a) (b)

Fig. 11 (a) Band structure of Si-deficient K8Ga8Si37. Fermi level is locatedat middle of bandgap. (b) Valence charge density for Si-deficientK8Ga8Si37 cut at plane including Si vacancy and its nearest neighborsites, calculated at highest occupied band. The contours are drawn withinthe range 0.001­0.1 e/¡3. The distance between successive contour linescorresponds to a factor of 1/10.

Fig. 10 Band structures of (a) K8Ga8Si38, (b) Rb8Ga8Si38 and (c) Cs8Ga8Si38. Fermi level was set at middle of band gap.

Energetic Stability and Thermoelectric Property of Alkali-Metal-Encapsulated Type-I Silicon-Clathrate from First-Principles Calculation 283

from experimental transport properties. They obtained a ZTvalue 0.185 for the single crystal,43) which is not consistentwith our prediction. However, the experimental latticethermal conductivity of Ba8Ga16Si30 was about 1W/mK atroom temperature,44) while that of Ba8Si46 we used was9.5W/mK. When we used the experimental lattice thermalconductivity,44) our predicted ZT value increased to 0.28.Note that the experimental condition was not necessarilyoptimized for carrier density. The Seebeck coefficient was¹42 µV, and the electrical conductivity was 3500 S/cmaccording to the estimation of Blake et al.43) When the carrierdensity was properly selected in our calculation, the abovequantities could be reproduced, and the resulting ZT valuewas 0.174, where the thermal conductivity was assumed to be1W/mK. This ZT value is close to the above estimated valueof 0.185.43)

Now, we will assume that the lattice thermal conductivityof all compositions was 1W/mK by the phonon scatteringof the Ga substitution, independent of temperature. Thisassumption can be justified because of weak temperaturedependence of lattice thermal conductivity of clathratematerial.5) Then, optimal carrier density can be determinedto maximize ZT because not only the electrical conductivitybut also the electrical thermal conductivity are carrier-relaxation-time dependent. The ZT values of each composi-tion after the re-optimization are represented as dotted linesin Fig. 11. The predicted ZT values are similar in electron-doped case, and about 0.66 at 900K. On the other hand, cleardistinction can be seen between X8Ga8Si38 and X8Ga8Si38(X = K, Rb, Cs) in hole-doped case. In the former com-position, ZT value at 900K is about 0.75. Consequently,K8Ga8Si38, Rb8Ga8Si38 and Cs8Ga8Si38 are proposed to bepromising p-type thermoelectric material.

In order to achieve ZT > 1 in those compositions, at least1.0 s is required for the carrier relaxation time. This longcarrier relaxation time is apparently unrealistic. Although ZTvalue of the other composition was predicted in our previousreport,45) no other composition was found to achieve ZT > 1.In addition, the carrier relaxation time is thought to becomeshorter with increasing temperature because of the phonon-electron coupling. Therefore, all Si clathrate ZT values at

900K should be lower than 1.0. The predicted ZT value of0.75 for K8Ga8Si38, Rb8Ga8Si38 and Cs8Ga8Si38 might be thereasonable upper limit for ZT at 900K.

However, present theoretical prediction does not excludethe possibility that ZT > 1 will be achieved in polycrystallineSi clathrate. Because the present calculation was performedfor single crystal, only the optimal chemical composition forachieving a high ZT value can be predicted. Maximizingthe ZT of the polycrystalline form, on the other hand, shouldbe more beneficial for real application. If the grain size issmaller than the mean free path of a heat carrying phonon andcarrier, the scattering of them is dependent on the grain size.At that small grain size limit, optimization of only the powerfactor is required, and ZT is expected to exhibit a maximumvalue.46) In order to verify this idea and our prediction,further studies on the experimental evaluation of thethermoelectric properties and analysis of the carrier andphonon scattering mechanisms must be needed.

4. Conclusion

In this study, possible combinations of alkali metal guestatoms and substitutional atoms in Type-I Si-clathrate wereinvestigated using first-principles calculations. From theformation energy, it was found that all alkali metals wereencapsulated as the guest element into a Si46 cage, and eitherAl or Ga was suitable as the substitutional atom. Variousconfigurations of Ga substitutional sites were evaluated, andthe (6c, 16i, 24k) = (6, 2, 0) configuration was the moststable for Li8Ga8Si38 and Na8Ga8Si38, while the (6c, 16i,24k) = (5, 0, 3) configuration was the most stable forK8Ga8Si38, Rb8Ga8Si38 and Cs8Ga8Si38. The different stableconfigurations of those two groups are suggested to originatein the degree of strain of the Si site in each cage. Fromthe formation energy, possible clathrate compositions wereproposed as K8Al8Si38, K8Ga8Si38, Rb8Al8Si38, Rb8Ga8Si38,Cs8Al8Si38 and Cs8Ga8Si38.

The thermoelectric properties were calculated from theBoltzmann transport equation using the band energiesobtained by the first-principles calculations. The optimalcarrier density for the Seebeck coefficient was dependent on

(a) (b)

Fig. 12 Predicted dimensionless figures of merit (ZT) for ternary Si clathrate. ZT for each composition was calculated using kl predictedfrom MD simulation.26,27)

K. Nakamura, S. Yamada and T. Ohnuma284

the band gap. Hole-doping gave similar electrical conductiv-ities for all clathrate compositions because the tops of thevalence bands are mainly composed of hybridized Ga­Si,and this was the same for all compositions. However, theorder of the electrical conductivity for electron-doping wasCs > Rb > K because of the changes in effective mass ofelectron. In addition, the large discrepancy between thetheoretically and experimentally determined band gaps of0.8 and 0.1 eV, respectively, for K8Ga8Si38 possibly originatein Si vacancy formation. The dimensionless figure of merit(ZT) for each composition was systematically predicted bycombining the Seebeck coefficient, electrical conductivity,and electronic thermal conductivity calculated in the presentstudy and the previously reported lattice thermal conduc-tivity. ZT µ 0.75 was predicted as the highest ZT value forhole-doped Cs8Ga8Si38.

Acknowledgments

K.N. appreciates Prof. Mizoguchi of The University ofTokyo for the discussion about the all-electron method.

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Energetic Stability and Thermoelectric Property of Alkali-Metal-Encapsulated Type-I Silicon-Clathrate from First-Principles Calculation 285