recent developments in thermoelectric...

Recent developments in thermoelectric materialsG Chen M S Dresselhaus G Dresselhaus J-P Fleurial and T Caillat

F FermindashDirac integral de ned by equation (8)f distribution functionEfficient solid state energy conversion based onograve Planckrsquos coeYcient divided by 2p J sthe Peltier effect for cooling and the Seebeck effect

for power generation calls for materials with high I current Aelectrical conductivity s high Seebeck coefficient k thermal conductivity W m ndash 1 K ndash 1S and low thermal conductivity k Identifying k

BBoltzmannrsquos constant J K ndash 1

materials with a high thermoelectric figure of merit L functions de ned by equation (4)Z(=S2sk) has proven to be an extremely m eVective mass kgchallenging task After 30 years of slow progress Q heat current Wthermoelectric materials research experienced a

S Seebeck coeYcient V K ndash 1resurgence inspired by the developments of new

T temperature Kconcepts and theories to engineer electron andV voltage Vphonon transport in both nanostructures and bulkv velocity m s ndash 1materials This review provides a critical summary

of some recent developments of new concepts and Z gure of merit K ndash 1

new materials In nanostructures quantum and ZT non-dimensional gure of meritclassical size effects provide opportunities to tailor

Q azimuthal anglethe electron and phonon transport throughh polar anglestructural engineering Quantum wellsL mean free path msuperlattices quantum wires and quantum dots

have been employed to change the band structure m chemical potential Jenergy levels and density of states of electrons j chemical potential divided by k

BT

and have led to improved energy conversion P Peltier coeYcient Vcapability of charged carriers compared to those of s electrical conductivity S mtheir bulk counterparts Interface reflection and the t relaxation time sscattering of phonons in these nanostructures v angular frequency Hz radhave been utilised to reduce the heat conductionloss Increases in the thermoelectric figure of meritbased on size effects for either electrons or Introductionphonons have been demonstrated In bulk

Solid state cooling and power generation based onmaterials new synthetic routes have led tothermoelectric eVects have been known since theengineered complex crystal structures with theSeebeck eVect (for power generation) and the Peltierdesired phononndashglass electronndashcrystal behavioureVect (for cooling and heat pumping) were discoveredRecent studies on new materials have shown that

dimensionless figure of merit (Z Ouml temperature) in the 1800s1 The Seebeck eVect is associated withvalues close to 1middot5 could be obtained at elevated the generation of a voltage along a conductor whentemperatures These results have led to intensified it is subjected to a temperature diVerence Chargedscientific efforts to identify design engineer and carriers (electrons or holes) diVuse from the hot sidecharacterise novel materials with a high potential to the cold side creating an internal electric eld thatfor achieving ZT much greater than 1 near room

opposes further diVusion The Seebeck coeYcient istemperature IMR397de ned as the voltage generated per degree of temper-ature diVerence between two pointscopy Dr Chen is in the Mechanical Engineering Department

Massachusetts Institute of Technology Cambridge MA02139 USA (gchen2mitedu) Dr M S Dresselhaus is S=shy

V1 2

DT1 2

(1)in the Department of Physics Department of ElectricalEngineering and Computer Science Massachusetts

The Peltier eVect re ects the fact that when carriersInstitute of Technology Cambridge MA 02139 USA Dr GDresselhaus is in the Francis Bitter Magnet Laboratory ow through a conductor they also carry heat TheMassachusetts Institute of Technology Cambridge MA heat current Q is proportional to the charge current I02139 USA Dr Fleurial and Dr Caillat are in the JetPropulsion Laboratory California Institute of Technology Q=PI (2)4800 Oak Grove Drive MS 277ndash207 Pasadena CA 91109

and the proportionality constant P is called theUSAPeltier coeYcient When two materials are joinedcopy 2003 IoM Communications Ltd and ASM Internationaltogether and a current is passed through the interfacePublished by Maney for the Institute of Materials Mineralsthere will be an excess or de ciency in the energy atand Mining and ASM Internationalthe junction because the two materials have diVerentPeltier coeYcients The excess energy is released tothe lattice at the junction causing heating and the

List of symbols de ciency in energy is supplied by the lattice creatingcooling The Seebeck and the Peltier coeYcients areC phonon volumetric speci c heat per unitrelated through the Kelvin relation P=ST where Tfrequency interval J m ndash 3 K ndash 1 Hz ndash 1

is the absolute temperature1 A typical thermoelectricE electron energy Je electron unit charge C cooler is shown in Fig 1a P-type and n-type semi-

DOI 101179095066003225010182 International Materials Reviews 2003 Vol 48 No 1 45

46 Chen et al Recent developments in thermoelectric materials

the units of inverse Kelvin and it often appears as aproduct with an absolute temperature T such as theaverage device temperature Thus the dimensionlessnumerical gure of merit Z T is often cited ratherthan Z by itself

The central issue in thermoelectrics materialsresearch is to increase Z T The best Z T materials arefound in heavily doped semiconductors Insulatorshave poor electrical conductivity Metals have rela-tively low Seebeck coeYcients In addition the ther-mal conductivity of a metal which is dominated byelectrons is in most cases proportional to the electri-cal conductivity as dictated by the WiedmannndashFranzlaw It is thus hard to realise high Z T in metals Insemiconductors the thermal conductivity has contri-butions from both electrons k

eand phonons k

p with

the majority usually coming from phonons Thephonon thermal conductivity can be reduced withoutcausing too much reduction in the electrical conduct-ivity A proven approach to reduce the phononthermal conductivity is through alloying2 The massdiVerence scattering in an alloy reduces the latticethermal conductivity signi cantly without muchdegradation to the electrical conductivity The com-mercial state of the art thermoelectric cooling mater-ials are based on alloys of Bi

2Te

3with Sb

2Te

3(such

as Bi0 middot5

Sb1 middot5

Te3 p-type) and Bi

2Te

3with Bi

2Se

3(such

as Bi2Te

2 middot7Se

0 middot3 n-type) each having a Z T at room

a cooler b power generator c actual device temperature approximately equal to 1 Refrigerators1 Illustration of thermoelectric devices based on such materials typically have a coeYcient

of performance (COP) of about 11 compared tocompressor based refrigerators with a COP between

conductor elements are interconnected on the cold 2 and 4 operating over a comparable working temper-and the hot sides such that a current ows through ature range Their low COP has limited thermo-all the elements in series while the energy they carry electric coolers to niche market sectors such as(by electrons and holes) leaves the cold side in parallel temperature stabilisation of semiconductor lasers andThermoelectric power generators work in reverse to picnic coolers The market for thermoelectric coolersthermoelectric coolers as shown in Fig 1b Because however is rapidly increasing partly due to thethe hot side has a higher temperature electrons and explosive growth of optical telecommunication Stateholes are driven to the cold side through diVusion of the art power generation materials are PbTe andand ow through an external load to do useful work Si

0 middot8Ge

0 middot2 which have been used in deep space radio-

Practical devices are made of many pairs of pndashn legs isotope thermoelectric power generators that operate(Fig 1c) usually arranged such that current ows in at ~900degC with a maximum eYciency of about 7series through all the legs and energy ows in parallel From equation (3) one can infer that the bestfrom the cold side to the hot side thermoelectric devices should have a thermal conduct-

In addition to the temperatures of the hot and cold ivity close to zero One possibility is using vacuumsides which are important to all thermal engines the between the cold and the hot side Electrons can beeYciency of actual thermoelectric devices is deter- emitted through a thermionic emission process frommined by the thermoelectric gure of merit a metal surface and ow through a vacuum This is

the principle behind thermionic power generatorsZ =

S2s

k (3) which were developed in the 1950s3 In a vacuum

based thermionic power generator the emitter is heldat a high temperature Electrons with energy higherwhere s is the electrical conductivity and k is the

thermal conductivity The appearance of S in Z is than the work function can escape from the emittersurface and reach the collector Conceivably ratherself-explanatory The reason that the electrical con-

ductivity s enters Z is due to Joule heating When a than for power generation vacuum thermionic emis-sion can also be used for cooling if a current drivescurrent passes through the thermoelectric elements

Joule heat is generated which can be conducted back electrons from the emitter to the collector as in avacuum tube The major problem however is thatto the cold junction The thermal conductivity k

appears in the denominator of Z because in thermo- most metals have a large work function value whichmakes room temperature refrigeration based onelectric coolers or power generators the thermoelec-

tric elements also act as the thermal insulation vacuum thermionic emission impractical4The progress since the 1960s in improving Z T hadbetween the hot and the cold sides A high thermal

conductivity causes too much heat leakage between been very slow before the 1990s The value of maxi-mum Z T had essentially remained around 1 andthe hot and the cold sides The gure of merit Z has

International Materials Reviews 2003 Vol 48 No 1

Chen et al Recent developments in thermoelectric materials 47

published three volume series1 3 A few other reviewsand introductory reports have also been publishedemphasising bulk thermoelectric materials1 4 ndash 1 7

Proceedings of International Conferences onlsquoThermoelectricsrsquo1 8 which are held annually and sev-eral volumes1 9 of the MRS proceedings on this topicgive broad coverage of progress made in recent yearsThis review is directed more towards low-dimensionalmaterials although bulk materials are also coveredto provide readers with a quick overview of currentthermoelectric materials research The selection of thecoverage was in uenced by the research focus ofthe present authors and re ects their assessmentof the eld The above mentioned references shouldbe consulted for more comprehensive coverage

Directions in search of high ZT2 Non-dimensional figure of merit ZT as function materialsof temperature for state of the art materials

Expressions for thermoelectric properties are oftenderived from the Boltzmann equation under the relax-

research funding in this area dwindled The landscape ation time approximation2 0

in thermoelectrics research changed quite signi cantlyin the 1990s due to several new conceptual devel- s=L( 0 ) k

e=

L( 2 )

e2 Tshy

L( 1 )L( 1 )

e2 TL( 0 ) S=shy

L( 1 )

eTL( 0 )(4)

opments and renewed interested from severalUS research funding agencies New developments whereoccurred both in bulk materials and in low-dimen-sional materials The best thermoelectric materials

L(a) =e2 Pd 3k

4p3 Ashy ccedil fF D

ccedil E B t[E (k)][v(k)]2 [E (k) shy m]awere succinctly summarised as lsquophononndashglass elec-tronndashcrystalrsquo (or PGEC in short ) which means that (5)the materials should have a low lattice thermal con-

In the above expressions fF D

is the FermindashDiracductivity as in a glass and a high electrical conduct-distribution t is the electron (hole) relaxation timeivity as in crystals5 In bulk materials the major newk is the electron wave vector k

eis the electronicconcept that was developed is the use of lsquophonon

contribution to thermal conductivity v is the electronrattlersrsquo to reduce the lattice thermal conductivitygroup velocity and e is the unit charge If the relax-These phonon rattlers are normally interstitial atomsation time is assumed to be a constant and a three-inserted into empty spaces in the host materials Theirdimensional (3D) parabolic electronic energy band isvibration is not in harmony with the atoms in theassumed Z T can be expressed as1 2 1host material thus scattering the phonons in the

original lattice In this connection several classes ofmaterials have been discovered andor re-investigated Z

3 DT =

[(5F3 2 3F

1 2) shy j]2 (3F

1 2 2 )

1B3 D+7F

5 2 2 shy (25F 23 2 6F

1 2)

(6)with regard to their potential for high Z T such asskutterudites and clathrates In low-dimensional wherematerials such as thin lms superlattices and quan-tum wires several approaches have been proposed B

3 D=

(m )3 2

3p2 A2kB

T

ograve 2 B3 2 k2BTm

ekp

(7)For transport along the lm plane (wire axis) direc-tion quantum size eVects are considered to increase

and m= (mxm

ym

z)1 3 is the eVective density of statesthe electronic power factor S2 s and boundary scat-

mass of electrons in the band kp

is the phonontering to reduce k6 7 For transport in the directioncontribution to the thermal conductivity k

Bisperpendicular to the lm plane several possibilities

Boltzmannrsquos constant m is the electron mobility jwere suggested One was to use the band-edge dis-is the chemical potential normalised by k

BT and F

iiscontinuity as a lter for cold electrons8 This was

the FermindashDirac integral de ned aslater developed into a thermionic emission coolingapproach9 1 0 Another approach was based on

Fi(j )= P

0

xi dx

exp (x shy j )+1 (8)phonon re ection at interfaces to reduce the lattice

thermal conductivity1 1 1 2 Figure 2 shows a lsquosnapshotrsquoof the reported Z T values The Z T values of low- In equations (6) and (7) the subscript 3D is used to

indicate that those expressions are derived consideringdimensional structures are subject to higher uncer-tainty and should be taken cautiously primarily due the density of states of 3D bulk crystals In low-

dimensional structures these expressions must beto the diYculties involved in characterising the Z Tof low-dimensional materials reformulated2 1 In equation (6) the reduced chemical

potential j is a free variable that can be controlledIn this paper it is intended to provide a concisecritical review of some recent developments in ther- by doping The optimum value for the chemical

potential is chosen to maximise Z T Therefore ther-moelectrics research An extensive review of most ofthe topics discussed here is contained in a recently moelectric materials development involves careful



control and optimisation of doping The only other weight material Its success relies on its multiplecarrier pockets that give a reasonably high density ofvariable that aVects the Z T value in equation (6) is

the B factor which depends on the electron eVective states and more importantly on the alloying methodthat signi cant lowers its thermal conductivity com-mass the carrier mobility and the phonon thermal

conductivity The larger the B factor the larger is Z T pared to that of bulk Si or Ge High temperatureoperation also helps to increase the Z T of Si

1 ndash xGe

xThus thermoelectric materials research is often

guided by nding materials that have a large B factor While the general Z T formulation for bulk mater-ials has played and will continue to play an instrumen-which include a large electron (hole) eVective mass

and a high mobility and a low lattice thermal con- tal role in developing strategies in the search forhighly eYcient thermoelectric materials it should beductivity Such materials are succinctly called

phononndashglass electronndashcrystal materials by Slack5 It kept in mind that these expressions are derived byusing a set of approximations Those related to theshould be pointed out that the requirements of a high

mobility (which needs a low mobility eVective mass) present discussion are given below1 Bulk density of states for electrons and holesand a high density of states (which demands a large

density of states eVective mass) are not necessarily Expressions such as (6) and (7) are derived byassuming 3D parabolic bands Quantum structuresmutually exclusive In anisotropic media either in

bulk form or in superlattices it is possible to have a to be discussed later have a drastically diVerentdensity of states and expressions (6) and (7) willsmall eVective mass in the current ow direction to

give a high mobility and large eVective masses in the change correspondingly2 Local equilibrium approximation Expressionsdirections perpendicular to the current ow to give a

high density of states for the transport coeYcients equations (4) and (5)are derived by assuming that electrons deviate onlyIt should be mentioned that the derivation of

equation (6) is based on the constant relaxation time slightly from their equilibrium distributions This isvalid only when the characteristic length along theapproximation A more realistic form of the relaxation

time has an energy dependence tEc which depends transport direction is much longer than the electronmean free path This assumption will not be valid foron the scattering mechanisms For example c=12

for optical phonons and c=shy 1 for acoustic transport at the interfaces and for carrier transportin the direction perpendicular to very thin lms Inphonons1 It can be shown that Z T also depends on

c Thus one strategy that is sometimes used to addition the electrons and phonons are typicallyassumed to be in thermal equilibrium This assump-improve Z T is to control the scattering mechanism

The thermal conductivity of phonons is also often tion is not necessarily true as in the well known hotelectron eVect in semiconductor electronicsmodelled from the Boltzmann equation under the

relaxation time approximation ie 3 Isotropic relaxation time for both electrons andphonons Many low-dimensional structures such assuperlattices are highly anisotropic Expressions such

kp= aelig

pP d 3k

8p3[v

p x(k)]2

ccedil fp

ccedil Tt

pograve v

as equation (9) for the thermal conductivity are nolonger correctMuch of the development in low-dimensional struc-

=1

3 P C(v)vp(v)L

p(v) dv (9) tures can be attributed to relaxing one or several of

these approximations This allows for more independ-ent control of S s or k as will be seen in thefor an isotropic bulk material where f

pis the phonon

distribution function C is the speci c heat of phonons following discussionat frequency v v

pis the phonon group velocity t

pis

the phonon relaxation time Lp

is the free path of Low-dimensional thermoelectricphonons at v and the summation is over the diVerent

materialsphonon polarisations To reduce the thermal conduct-ivity materials with a small phonon group velocity Low-dimensional materials such as quantum wells

superlattices quantum wires and quantum dots oVerand a short relaxation time are desired Roughlyspeaking the phonon group velocity is proportional new ways to manipulate the electron and phonon

properties of a given material In the regime whereto (K m)1 2 where K is the spring constant betweenthe atoms and m is the mass of the atom Thus quantum eVects are dominant the energy spectra of

electrons and phonons can be controlled throughmaterials with high atomic mass are often used forthermoelectric materials The phonon relaxation time altering the size of the structures leading to new ways

to increase Z T In this regime the low-dimensionalcan be reduced by scattering such as throughalloying2 and by adding phonon rattlers1 5 structures can be considered to be new materials

despite the fact that they are made of the same atomicSuccessful bulk thermoelectric materials that weredeveloped in the past were directed by the principles structures as their parent materials Each set of size

parameters provides a lsquonewrsquo material that can bederived from the above general discussion for bulkmaterials For example Bi

2Te

3was tested because of examined to a certain extent both theoretically and

experimentally in terms of its thermoelectric prop-its high atomic weight2 2 Other important character-istics of Bi

2Te

3that were discovered later such as its erties Thus searching for high Z T systems in low-

dimensional structures can be regarded as the equival-multiple carrier pockets high mobility and low ther-mal conductivity all contributed to its high Z T value ent of synthesising many diVerent bulk materials and

measuring their thermoelectric properties Becausebecause all these factors work favourably to increasethe B factor In contrast SiGe is not a high atomic the constituent parent materials of low-dimensional



structures are typically simple materials with wellknown properties the low-dimensional structures areamenable to a certain degree of analysis predictionand optimisation In contrast theoretical predictionsfor bulk materials properties are often diYcult2 3 andthe investigation of each new material presents adiVerent set of experimental and theoretical chal-lenges When quantum size eVects are not dominantit is still possible to utilise classical size eVects to alterthe transport processes as for example the exploi-tation of boundaries to scatter phonons more eVec-tively than electrons Investigations over the pastdecade on low-dimensional structures have exploitedboth quantum and classical size eVects for electronsand phonons The focus of discussion here is along

3 Calculated dependence of non-dimensionalthree directions One is for electron transport parallelfigure of merit ZT (within quantum well orto the plane of thin lms or along the axis ofwithin quantum wire) on well or wire width fornanowires Another is for the electron transport per-Bi2Te3-like material at optimum doping

pendicular to the lm plane The third emphasises concentration for transport in highest mobilityphonon transport Although no devices based on low- direction also shown is ZT for bulk (3D) Bi2 Te3dimensional structures are close to commercialisation calculated using corresponding 3D model625

research on low-dimensional thermoelectricity hasbeen stimulating to recent developments in thermo-electrics research and new research groups worldwide 4 and that the model calculations were in good

agreement with the experimental observations Soonare now starting to bring new ideas to the eld basedon nanostructures thereafter it was shown that an enhanced thermoelec-

tric gure of merit could also be observed for p-typePbTe quantum wells3 0 3 1 which is of course anElectron band gap engineered materialsimportant consideration for thermoelectric devicestransport parallel to film plane or nanowirewhich depend on having both n-type and p-type legsaxis

These modelling calculations and experimentalThe development of band gap engineering in quantum

data are based on the consideration of transportstructures for thermoelectric applications started with inside the quantum wells only and thus the reportedthe modelling of single quantum wells6 and soon Z T values are referred to as (Z T )

2 D Theoretical

moved on to the consideration of superlattices2 1 Itmodelling pointed out that when the barrier regions

was soon recognised that quantum wires would oVer of the superlattices are also considered the overallmore quantum con nement and therefore would have

gain in Z T of the whole structure is considerablyadvantages over quantum wells for thermoelectric

reduced3 2 3 3 because the barriers do not contributeapplications2 4 2 5 as shown in Fig 3 In due course to the electrical transport but do contribute to theresearchers also began to consider the bene ts of

reverse heat conduction In the limit that the barrierquantum dot systems for thermoelectric applications

is thin it was also suggested that tunnelling betweenThe key idea is to use quantum size eVects to increase quantum wells reduces the Z T enhancement insidethe electron density of states at the Fermi level and

an individual quantum well How to address thisin this way to optimise the power factor Further

issue is very important for the utilisation of quantumbene ts to thermoelectric performance can be realised size eVects on the electron transport along the lmby exploiting boundary scattering to reduce the ther-

plane One possible approach suggested by themal conductivity preferentially without much loss to

Dresselhaus group is to use both the quantum wellsthe electrical conductivity and barriers for thermoelectric transport along the

lm plane3 4 For example by proper choice of theQuantum wells and superlatticesThe most elementary generic calculations6 2 4 2 5 and widths of the quantum wells and barriers for

GaAsAlAs superlattices the carriers in the C-pointalso more sophisticated calculations on speci c mater-ials2 6 both predict enhanced thermoelectric perform- and L-point valleys contribute to transport in the

GaAs regions while the X-point valleys can contrib-ance within the quantum wells of multiwellsuperlattices relative to bulk materials of the same ute to transport in the AlAs regions3 5 Furthermore

for a given superlattice system such as SiGe superlat-stoichiometry To show proof of principle experimen-tally special superlattices were designed by the tices and their alloys3 6 model calculations can be

used to optimise the superlattice geometry to achieveDresselhaus group and fabricated by the Harmangroup at MIT Lincoln lab for PbTe based superlat- the maximum Z T of the whole superlattice structure

Although there are limited experimental data suggest-tices2 7 and by the Wang group at UCLA to makeSiGe superlattices2 8 2 9 The rst proof of principles ing this possibility more experimental eVorts are

needed to demonstrate the eVectiveness of this carrierexperiment to con rm the enhancement of Z T withina quantum well was reported for n-type PbTe quan- pocket engineering approach

The theoretical work has inspired experimentaltum wells within PbTeEuxPb

1 ndash xTe superlattices2 7

The results showed that the power factor within the studies of thermoelectric eVects in superlattices Thepast few years have seen steady improvements in thePbTe quantum well could be increased by a factor of



5 Cross-sectional view of Bi nanowires incylindrical channels of 65 nm average diameter4 Thermoelectric non-dimensional figure of meritwithin anodic alumina template shown asversus carrier concentration for high qualitytransmission electron microscope (TEM) imagebulk PbTe material in comparison to muchtemplate has been mostly filled with Bi andhigher ZT for PbSeTe PbTe quantum dotTEM image was taken after top and bottomsuperlattice structure at 300 K7 (courtesy Tsides of sample had been ion milled with 6 kVHarman)Ar ions4448

thermoelectric performance of speci c superlatticesnels from 7 to 100 nm in diameter and 50 mm in

For example experimental thermoelectric data fromlength with packing densities of 101 0 channels cm ndash 2

Harmanrsquos group demonstrated that for PbTeTe(Refs 44 45) These pores within the channels can

superlattices obtained by the addition of a few nano-then be lled with promising thermoelectric materials

metres of Te above the PbTe layer before the barrier such as Bi or Bi1 ndash x

Sbx

alloys Because of the lowlayer is added the Z T increased from 0middot37 to 0middot52 at

eVective masses and large anisotropy of the constantroom temperature and this increase in Z T was associ-

energy surfaces of these materials quantum con ne-ated with the formation of a quantum dot structure ment eVects at 77 K are predicted for wire diametersat the interface3 1 The Harman group further disco-

as large as 50 nm4 6 as discussed further below Thesevered experimentally that quantum dot superlattices

eVects have been veri ed experimentally by transportbased on PbTePbSeTe have an even higher Z T For measurements4 4 4 7example PbSeTePbTe quantum dot superlattices

Figure 5 shows an example of an anodised aluminashow a large enhancement in the Seebeck coeYcient

template which is lled with Bi using the pressurerelative to bulk PbTe samples with the same carrier injection method4 8 One important advantage of theseconcentration7 At this stage detailed mechanisms for

Bi nanowires is their crystal properties which can bethe reported Seebeck coeYcient enhancements in

seen in the X-ray and electron diVraction patternsthese quantum dot superlattices are not clear Among shown in Fig 6 for an array of Bi nanowires 52 nmthe speculations are both quantum con nement eVects

in average diameter indicating that each wire has aand a more favourable scattering mechanism associ-

similar crystal orientation along the nanowire axisated with quantum dots Furthermore an estimation This is veri ed in Fig 6 by comparison of the X-rayof Z T for such quantum dot superlattices based on

diVraction pattern with the corresponding electronthe phonon thermal conductivity of equivalent alloys

diVraction pattern for a typical nanowire showing athat are considered by the authors as conservative unidirectional orientation of the nanowires along theyields a room temperature value of 0middot9 which is more

wire axes In principle the crystalline orientation canthan a factor of 2 greater than has been achieved

be controlled but thus far the preferred growthwith the best bulk PbTe material (see Fig 4) EVorts direction for a given growth method has dominatedare presently underway by several groups worldwide

the nanowire crystalline orientation As a result ofto obtain reliable measurements of the in-plane ther-

the high crystallinity of these nanowires high carriermal conductivity of small superlattice samples3 7 ndash 4 1

mobility and long mean free paths are achievableThe growth by molecular beam expitaxy of PbTe

which is of direct bene t for thermoelectricbased superlattices and quantum dot superlattices

applicationscan be relatively fast making it feasible to grow very In addition to pressure injection physical vapourthick superlattices (50ndash100 mm total thickness) for

deposition4 9 has also been used to ll the templatessimple device testing4 2

and to make Bi nanowires with single crystal prop-erties having the same crystal structure and latticeQuantum wires

General theoretical considerations suggest that constants as bulk Bi Electrodeposition is very attract-ive for lling the pores of an alumina templatebecause of their increased quantum con nement

eVects 1D quantum wires could have an even larger because of the ease in making good electrical contactsThe exibility of the electrodeposition approach hasenhancement in Z T 2 4 4 3 than 2D quantum wells (see

Fig 3) To fabricate controlled arrays of quantum allowed nanowire arrays of Bi CoSb3 and Bi

2Te

3to

be produced The nanowire arrays thus far producedwires anodic alumina (Al2O

3) templates have been

developed and these templates can be made to have by electrodeposition are polycrystalline and are there-fore expected to have lower carrier mobilities thanregular triangular arrays of porous nanoscale chan-



6 X-ray diffraction pattern for anodic alumina Binanowire composites (average wire diameter ofBi nanowires is about 52 nm) and selectedarea electron diffraction pattern (inset) takenfrom same sample these two experimentalresults indicate that Bi nanowires are highlycrystalline and possess strongly preferredgrowth orientation4445

the arrays prepared by lling from the vapour phaseor by pressure injection

Bismuth has a highly anisotropic band structurewith some very small eVective mass tensor com-ponents (~10 ndash 2 free electron masses) and high a subband structure at 77 K of Bi quantum wires orientedmobility carriers making it a very attractive thermo- along [011Acirc 2] growth direction showing energies of highest

subbands for T-point hole carrier pocket L-point electronelectric material except that in bulk form Bi is apockets (A B and C) as well as L-point holes which aresemimetal with equal concentrations of electrons andindicated in Brillouin zone shown on right (zero energyholes thus leading to a nearly complete cancellation refers to conduction band edge in bulk Bi) as wire diameter

between the positive and negative contributions to decreases conduction subbands move up in energy whilethe Seebeck coeYcient However for Bi as a quantum valence subbands move down at dc=49middot0 nm lowest

conduction subband edge formed by L(BC) electronswell or a quantum wire the band edge for the lowestcrosses highest T-point valence subband edge andsubband in the conduction band rises above that forsemimetalndashsemiconductor transition occurs (D0 is band

the highest subband in the valence band (see Fig 7) overlap and EgL is L-point band gap)45 b Fermi surfaces ofthereby leading to a semimetalndashsemiconductor trans- Bi shown in relation to Brillouin zone corresponding to

fifth-band hole pocket about T-point and three sixth-bandition This is predicted to occur at a wire diameter ofL-point electron pockets labelled A B and C mirror plane49 nm for Bi nanowires oriented along the favouredsymmetry of bulk bismuth structure results ingrowth direction Bismuth nanowires are of interest crystallographic equivalence of L-point carrier pockets B

for thermoelectric applications when in the semicond- and C however L-point carrier pocket A is not equivalentucting regime under heavy doping conditions crystallographically to carrier pockets B or C

Temperature dependent resistance measurements 7 Diameter dependence of band edges for Binanowires 3D Brillouin zone in b shows(see Fig 8a) in conjunction with model calculationsimportant band edges participating in semi-(see Fig 8b) show that these bismuth nanowires aremetalndashsemiconductor transitionconverted from a semimetal into a semiconductor due

to quantum size eVects5 0 This is seen by the mono-tonic temperature dependence of the resistance in the the resistance of both nanowire arrays and of single

nanowires These diYculties arise from problems withsemiconducting phase for wire diameters below 48 nmand the non-monotonic behaviour with temperature making good ohmic contacts to all the nanowires of

a nanowire array and with the oxidation of thefor wires with diameters larger than 70 nm abovewhich the wires are in the semimetallic regimes in individual nanowires when they are removed from

their templatesagreement with theoretical predictions (Fig 8b) Thenon-monotonic dependency of resistance on the The Bi

1 ndash xSb

xalloy system is interesting as a

method for obtaining a low-dimensional p-type semi-nanowire diameter is due to the extrinsic carrierscontributed by uncontrollable impurities which conductor with very low eVective masses and high

mobility carriers which is important because thermo-becomes more important as band gap increases Sofar there exists no conclusive experimental data show- electric devices require both n-type and p-type legs

Since Sb is isoelectronic with Bi and has the sameing an enhanced power factor in Bi nanowire arraysdue to diYculties in measuring the absolute value of A15 crystal structure as Bi bulk Bi

1 ndash xSb

xalloys



SM=regions where alloy is semimetal SC=region wherealloy is semiconductor

9 Schematic diagram for energy bands near Fermilevel for Bi1 ndash xSbx bulk alloys as function of x atlow T ( agrave 77 K)51 when highest valence band isat L-point direct gap semiconductor results forx values where highest valence states are at Tor H indirect semiconductor is obtained

a experimental temperature dependence of normalisedresistance for Bi nanowire arrays of various wire diametersprepared by vapour deposition method in comparison withcorresponding data for bulk Bi measurement of resistance 10 Phase diagram of electronic band structure ofwas made while Bi nanowires were in their alumina Bi1 ndash xSbx nanowires52 bold arrow in centretemplates using two-probe measurement technique49

indicates condition where 10 hole pocketsb calculated temperature dependence of resistance for Bi(about T-point 3 L-points and 6 H-points innanowires of 36 and 70 nm using semiclassical transportBrillouin zone) coalesce in energymodel45

8 Temperature dependence of resistancemeasurements for Bi nanowires concentration the wires are predicted to be semimet-

allic a direct gap semiconductor or an indirect gapsemiconductor The calculations of Fig 10 show howprovide a high mobility material but with electronic

properties that can be varied considerably as the Sb variation of the nanowire diameter can be used tochange the electronic structure quite dramaticallyconcentration is varied (see Fig 9) Of particular

interest for thermoelectric applications is the low with no basic change occurring in the crystal struc-tures Of particular interest in this diagram for theSb concentration range (below x =0middot07) where the

bulk material is semimetallic the regions for Bi1 ndash x

Sbx

nanowires is the point at x =0middot13 and awire diameter of 60 nm where the L-point T-point0middot07ltx lt0middot09 and 0middot16ltx lt0middot22 where bulk

Bi1 ndash x

Sbx

is an indirect gap semiconductor and nally and H-point hole subband edges are all degeneratewith one another leading to a very large density ofthe region 0middot09ltx lt0middot16 where bulk Bi

1 ndash xSb

xis a

direct gap semiconductor5 1 Calculations for the elec- hole states Such a system where all 10 hole subbandedges are degenerate in energy is predicted to exhibittronic structure for Bi

1 ndash xSb

xnanowires (see Fig 10)5 2

show that variation of the wire diameter leads to an enhanced Seebeck coeYcient and an increasedZ T5 2 5 3 The feasibility of observing interesting trans-another important degree of freedom in the control

of the electronic structure showing regions where port properties in Bi1 ndash x

Sbx

nanowires has beendemonstrated by showing that the high degree ofdepending on the nanowire diameter and Sb alloy



crystallinity of the Bi nanowires is preserved upon Sb Bowers predicted that solid state thermionic coolershave a larger cooling power than that of thermoelec-alloying as is the high carrier mobility

Thus far progress in the fabrication and measure- tric devices Mahan and co-workers5 7 5 8 followed upwith a thermionic cooling and power generationment of the thermoelectric performance of test

samples is most advanced for 2D systems where model for multiple layer structures In this case theadvantage of the non-uniform heat generation insidesuperior performance has already been demonstrated

The excellent Z T3 D

for these superlattices is attributed a double heterojunction structure9 due to the ballisticelectron transport is lost Their initial calculation5 7 5 8to the presence of quantum dot structures at the

nanoscale7 whose behaviour is not yet understood in suggests that multilayer thermionic coolers can havea high eYciency Yet in another paper Vining anddetail nor has the structure yet been optimised for

thermoelectric performance The 1D quantum wires Mahan5 9 compared the equivalent B factor that deter-mines the Z T for thermionic multiple layer structureshave in contrast been modelled in detail though

materials problems involving surface oxidation and and similar thermoelectric multiple layer structuresand found that the B factor of thermionic structures iswith the formation of good individual contacts to the

nanowires have slowed the experimental evaluation not larger than that for thermoelectric structuresTheir conclusion is that the thermal conductivityand optimisation of the thermoelectric performance

of the quantum wires It is thus expected that further reduction1 1 1 2 will be the major bene t of multiplelayer structures which is consistent with the studiesemphasis will be given to materials science issues

before the construction of reliable quantum wire carried out by the Ehrenreich group5 5 In addition toan all solid state cooling strategy vacuum thermionicthermoelectric devices will be seriously undertakenrefrigeration based on resonant tunnelling throughtriangular wells and small spatial gaps in the vacuum

Electron transport perpendicular to film plane has been proposed recently6 0 6 1 Theoretical calcu-lations predict operation at room temperature andMost of the eVort so far has been focused on thin

lms and superlattices There are two lines of consid- below with a cooling power density of 100 W cm ndash 2 No net cooling based on such vacuum thermioniceration for electron transport perpendicular to the

thin lm plane (i ) control of the density of electron coolers has been reported experimentallyTheoretically the modelling of electron transportstates using quantum size eVects and (ii ) energy

ltering through thermionic transport Two theo- perpendicular to the con nement direction is con-siderably more diYcult compared to that along theretical papers considered quantum size eVects

on thermoelectric properties for electron transport lm plane because the lm thickness may be compar-able to several characteristic lengths of the chargeperpendicular to the superlattice plane5 4 5 5 A slight

increase in Z T for SiSiGe superlattices made of carriers including the wavelength and the mean freepath Thus far quantum based models which considerextremely thin layers (~5 A) is predicted5 4 Radtke

et al5 5 studied HgxCd

1 ndash xTe superlattices and their the electrons as totally coherent and thermionic

emission models which consider the electrons ascalculation shows that a 20 increase in power factoris possible in narrow-well narrow-barrier superlattice totally incoherent have been constructed But there

is no theory so far for the overlapping region Insystems but suggested that the gain in Z T will mostlikely come from a thermal conductivity reduction addition there are also thermoelectric eVects inside

the lm which may be coupled with thermionic eVectsrather than from a power factor increase So far thereseems to be no experimental eVort aimed at pursuing at the interface to yield a total Z T of the structure

There have been a few studies that treat both eVectsa power factor increase due to quantum size eVectsfor transport perpendicular to superlattice planes and phonon size eVects by Zeng and Chen6 2 ndash 6 4 Their

modelling suggests that when the lm is very thinIn the limit that the quantum size eVect is notimportant for this case there is still a possibility for energy conversion is dominated by the thermionic

emission and when the lm is thick thermoelectricincreasing Z T It was proposed that the energy bar-riers at the junctions of diVerent materials be used as transport governs the energy conversion eYciency In

the intermediate lm thickness range both eVectsan energy lter to increase the thermoelectric energyconversion eYciency8 Electron transport over such contribute to the nal Z T Their work also suggests

that the thermal conductivity reduction will contrib-barriers is described by thermionic emission theoryIn a theoretical paper Mahan5 6 considered the cool- ute more to the Z T enhancement compared to the

thermionic emission Z T which is consistent withing power of vacuum based thermionic coolers whichare based on the same principle as vacuum based previous studies that inferred the importance of the

thermal conductivity reduction based on experimentalthermionic power generators His calculation showedthat vacuum based thermionic coolers will not be results5 5 5 9

Experimentally Shakouri and co-workers haveable to run at room temperature because of the largework function of known materials and because of fabricated thin lm thermoelectric coolers based on

single heterojunction structures6 5 and superlatticespace charge eVects Shakouri and Bowers9 suggestedthat this could be circumvented using double hetero- structures6 6 ndash 7 0 based on InP and SiGe superlattices

The maximum temperature rise measured on a singlejunction structures The barrier height between twomaterials can be precisely tailored in theory as well element device is 12 K at a 200degC substrate temper-

ature from SiGe superlattices7 0 Mahan1 0 pursuedas in practice for certain materials systems Anotheradvantage of this approach is that Joule heating is metalndashsemiconductor superlattice structures for cool-

ing No cooling eVect has yet been reported frommostly rejected at the hot side due to the ballistictransport Based on a simpli ed model Shakouri and such structures It should be emphasised that testing



a single device is very diYcult and involves many removal of the substrate3 7 ndash 3 9 8 3 Very few studies havereported thermal conductivity in both the in-planeforms of losses that may degrade the device perform-

ance Thus these devices may have better performance and cross-plane directions3 8 4 0 All these experimentscon rmed that the thermal conductivities of the super-if successfully developed as arrays rather than as

individual devices So far there are no systematic lattices in both directions are signi cantly lower thanthe predictions based on the Fourier law and theexperimental investigations on whether the observed

cooling is due to thermionic or thermoelectric mech- properties of their bulk parent materials In the cross-plane direction the thermal conductivity values cananisms or both Field emission coolers based on

GaAs are also being investigated but no cooling de nitely be reduced below that of their correspond-ing alloys1 2 8 1 In the in-plane direction the reductioneVect has been observed so far7 1

is generally above or comparable to that of theirEngineering phonon transport equivalent alloys4 0 although a few experimental data

indicate that k values lower than those of the corres-Phonon transport in low-dimensional structures isalso aVected by size eVects and can be utilised to ponding alloys are possible1 1

A few groups have developed theoretical expla-increase Z T Size eVects in the thermal conductivityare a well known phenomenon that is important at nations for the thermal conductivity of superlattices

for both in-plane and cross-plane directions Twolow temperatures for bulk materials7 2 Several studiesof the thermal conductivity of thin lms were carried schools of thought are apparent from the literature

One starts from the phonon spectrum calculation andout in the 1970s and 1980s mostly for polycrystallinemetallic or semiconductor thin lms The rst experi- attributes the thermal conductivity reduction to

changes in the group velocity density of states andment on superlattices was performed by Yao3 7 for thethermal conductivity along the lm plane He scattering mechanics8 8 ndash 9 3 The other approach starts

from the simple picture of interface re ection andobserved that the thermal conductivity of the super-lattices investigated was higher than that for their treats phonon transport in terms of particles9 4 ndash 9 7 The

former assumes that phonons are totally coherentcompositionally equivalent alloys One can easilyinfer that the reported values are also signi cantly and the latter treats phonons in each layer as totally

incoherent The coherent phonon picture is accuratelower than the values calculated from bulk propertiesaccording to the Fourier theory The rst experiment if the interfaces and internal scattering do not destroy

the coherence of the phonons Compared to thein the cross-plane direction was reported by Chenet al3 8 They measured the thermal conductivity of a coherent picture the particle approach does not treat

the following mechanisms correctly (i) phonon inter-semiconductor laser structure which contained shortperiod superlattices in both directions and observed ference which gives minigaps in the superlattice

phonon spectrum (ii) phonon tunnelling whicha factor of 10 reduction of k in the cross-planedirection compared to predictions by the Fourier occurs for very thin layers above the critical angle for

total internal re ection of phonons and (iii ) longtheory The reduction in the in-plane direction issmaller but also signi cant In a review paper Tien wavelength phonons which do not lsquoseersquo the existence

of the interfaces These three factors however do notand Chen7 3 suggested that the new spectrum insuperlattices can potentially lead to super thermal seem to be dominant in the observed thermal conduct-

ivity behaviour of superlattices This is because heatinsulators The studies by Yao3 7 and Chen andco-workers3 8 3 9 were mainly geared towards thermal conduction involves the contribution from all allow-

able phonons covering the entire phonon frequencymanagement applications for semiconductor lasers7 4

Venkatasubramanian proposed to use the potentially range Minigaps created by interference eVects coveronly a small fraction of the total thermal energylow cross-plane thermal conductivity of superlattices

for thermoelectric devices1 1 7 5 The idea is to use the Tunnelling is important only when each layer is only1ndash3 monolayers thick due to the small phonon wave-phonon re ection at interfaces to reduce the thermal

conductivity while maintaining the electron trans- length In addition the diVuse interface scatteringoccurring at most interfaces which seems to be a verymission at the interfaces by combining materials with

small or ideally zero band-edge oVset Such struc- important factor destroys the phonon coherenceComparison of lattice dynamics9 2 acoustic wavetures are called electron-transmitting phonon-block-

ing structures This strategy on the Bi2Te

3Sb

2Te

3propagation9 8 and Boltzmann equation9 4 9 6 9 7 simu-lations with experimental data by Chen andsystem seems to have led to a signi cant increase of

Z T 1 1 7 6 as indicated in Fig 2 As a word of caution co-workers leads to the conclusion that the majorreason for the observed thermal conductivitybecause the characterisation of Z T is extremely

diYcult in this direction the values should be sup- reduction in the cross-plane direction is the phononre ection particularly the total internal re ection1 2ported by more research preferably by diVerent

groups Although phonon con nement due to the spectralmismatch can potentially contribute signi cantly toExtensive experimental data on the thermal con-

ductivity of various superlatttices including Bi2Te

3 the thermal conductivity reduction it is likely that

many phonons leak out due to inelastic scatteringSb2Te

37 5 7 7 78 GaAsAlAs37 ndash 39 7 9 8 0 SiGe40 8 1 ndash 8 3 InAs

AlSb8 4 InPInGaAs8 5 CoSb3IrSb

38 6 and PbTe For both the in-plane and the cross-plane directions

diVuse interface scattering of phonons seems to playbased superlattice4 1 have been reported in recentyears Most of these measurements are in the cross- a crucial role As an example in Fig 11a the simulated

thermal conductivity reduction in SiGe superlatticesplane direction7 5 8 1 ndash 8 6 using the 3v method8 7 or theoptical pump-and-probe method8 0 Measurements in both the in-plane and the cross-plane directions

based on lattice dynamics modelling of the phononalong the lm plane direction relied heavily on the



a lattice dynamics simulation of thermal conductivity normalised to bulk values92 b temperature dependence of thermalconductivity of SiGe (2020 A) superlattice along (subscript x) and perpendicular (subscript z) to film plane40 c thicknessdependence of thermal conductivity of GaAsAlAs superlattices along film plane (GaAs and AlAs layers are of equal layerthickness)94 d period thickness dependence of thermal conductivity of SiGe superlattices in cross-plane direction97 (alsoplotted in bndashd are fittings based on solutions of Boltzmann equations with p representing fraction of specularly scatteredphonons at interface and values calculated based on Fourier law and measured bulk thermal conductivities of Si and Ge)

11 Results indicating crucial role of diffuse interface scattering of phonons

spectrum in superlattices9 2 is shown in comparison is contrary to most experimental observations onGaAsAlAs superlattices and SiGe superlatticesto experimental results (Fig 11bndash11d )4 0 on a SiGe

superlattice The simulation based on the phonon as shown in Fig 11c and 11d9 4 9 7 except in the verythin layer region where experimental data ongroup velocity reduction leads to only a small

reduction in k in the in-plane direction while the Bi2Te

3Sb

2Te

3superlattices indeed show a recovery

trend with decreasing period thickness1 1 Acousticcross-plane direction shows a relatively large drop(Fig 11a) The lattice dynamics models imply perfect wave simulation suggests that the recovery in the

thermal conductivity at the very thin period limit isinterfaces and thus no diVuse interface scatteringExperimentally the thermal conductivity reductions due to the tunnelling of phonons from one layer into

another for incidence above the critical angle9 8 Inin both directions are much larger as shown inFig 11b In addition the lattice dynamics simulation reality however there also exists the possibility that

such a recovery is due to interface mixing that createsleads to a thermal conductivity that rst decreaseswith increasing superlattice period thickness and then alloys rather than superlattices In the thicker period

regime the Boltzmann equation based modelling thatapproaches a constant that is signi cantly lower thanthat of its corresponding bulk values Such behaviour treats phonons as particles experiencing partially



specular and partially diVuse scattering at the He proposed that the correct starting point shouldbe from the following expression for the thermalinterfaces can lead to a reasonable t to the experi-

mental data as shown in Fig 11bndash11d The parameter conductivityp in Fig 11bndash11d represents the fraction of specularlyscattered phonons at the interface and (1 shy p) that of

kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0

C(v)n(vhQ)the diVuse scattering that may be caused by interfacemixing and roughness or anharmonic forces at theinterface The agreement between modelling and

Ouml L(vhQ) cos2 h sin h dhDH dv (10)experimental results suggests that the phonon coher-ence length in superlattices is short and the loss ofcoherence is probably due to diVuse interface where h and Q are the polar and azimuthal angles

formed with the heat ux direction The task ofscatteringTo exhibit signi cant size eVects the phonon mean reducing the thermal conductivity is to reduce the

value of the above integral Low-dimensional struc-free path in the bulk material should be larger thanthe lm thickness or other characteristic lengths of tures oVer several new ways to reduce the thermal

conductivity integral in equation (10) First the groupthe structure The estimation of the phonon meanfree path however must be done carefully Often velocity can be altered in nanostructures The forma-

tion of standing waves in nanostructures means thatone tends to estimate the phonon mean free path Lfrom the simple kinetic formula k =CvL3 using the the group velocity becomes smaller thus reducing the

thermal conductivity In superlattices the bulk acous-speci c heat and speed of sound in bulk materialsThe mean free paths of phonons that actually carry tic phonons can be changed into optical phonons

thereby drastically reducing their group velocitythe heat could be much longer because (i ) opticalphonons contribute to the speci c heat but not much Second it is possible to induce anisotropic scattering

in low-dimensional structures For example interfaceto thermal conductivity due to their small groupvelocity and (ii) the acoustic phonon group velocity re ection and transmission are highly angle depen-

dent Total internal re ection means that phononscan be much smaller than the speed of sound due todispersion eVects For example a careful estimation above the critical angle will be re ected backwards

As another example the optical phonons in twoof the phonon mean free path in silicon leads to2500ndash3000 A9 7 9 9 compared to the kinetic theory materials have totally diVerent frequencies It is likely

that the scattering of optical phonons at the interfacevalue of ~400 A Because of this some apparentlylow thermal conductivity materials such as will be highly directional ie the optical phonons will

be preferentially scattered backwards In the contextBi2Te

3Sb

2Te

3and CoSb

3IrSb

3superlattices can

actually be engineered to have lower values by explor- of the phonon dispersion curve this is called phononcon nement The eVects of total internal re ectioning size eVects Another important point is that typi-

cally the thermal conductivity reduction in the cross- and the phonon con nement on the thermal conduct-ivity can also be interpreted as reducing the angleplane direction is larger than in the in-plane direction

because interfaces impede the phonon transport more and frequency integration limits of equation (10)thus decreasing the thermal conductivity Third thein the cross-plane direction than along the lm plane

as is suggested by Fig 11b For polycrystalline mater- speci c heat of nanostructures can be changed bychanging (i ) the density of states and (ii ) the degreesials this may not necessarily be true because the

columnar grain structures can actually cause a more of freedom of the atomic vibrations Theoretical stud-ies on superlattices however suggest that thesesigni cant reduction in the in-plane direction as is

observed in diamond thin lms1 0 0 Other factors such changes are not strong except at low temper-atures1 0 3 ndash 1 0 5 Based on these arguments Chen1 2 sug-as the dislocation orientation (occurring in threading

dislocations) may also create more scattering in the gested that low-dimensional structures may have asmaller minimum thermal conductivity Figure 12in-plane direction although there have been neither

detailed theoretical nor experimental studies shows the experimental thermal conductivity of aSiGe superlattice compared to predictions based onExisting experimental data have clearly shown that

the thermal conductivity of superlattices can be made the minimum theoretical thermal conductivity of bulkSi and Ge The gure suggests the possibility ofsmaller than that of their corresponding alloys

Remember that alloying has been used in thermoelec- reaching values lower than that theoretically attain-able in bulk materialstric materials research as an eVective way to reduce

the lattice thermal conductivity This raises the ques- Thus it seems that the following strategies may bepursued to engineer the phonon transport in ordertion of what is the minimum thermal conductivity of

superlattices Slack1 0 1 proposed that the minimum to reduce the lattice thermal conductivity1 For transport along the interfaces ie along thethermal conductivity one can reach for a material is

when the phonon mean free path in equation (9) is lm plane and wire axis the thermal conductivity canbe reduced by creating diVuse interface scattering andreplaced by the wavelength Later Cahill et al1 0 2

further limited it to half the wavelength The more reducing the interface separation distance In additionto the naturally existing interface roughness due tofundamental question is whether low-dimensional

structures are subject to the same limit or not Chen1 2 the mixing of atoms at the interfaces other possibil-ities are arti cially corrugated interfaces such as thinargued that the same minimum may not be applicable

to low-dimensional materials which are highly aniso- lms grown on step-covered substrates and quantumdot interfaces In controlling the interface structurestropic because anistropy causes directional depen-

dence of the relaxation time and of the group velocity for phonon thermal conductivity reduction the



than periodic superlattices or composite superlatticeswith diVerent periodicities may provide methods tolocalise some if not all of these long wavelengthphonons

4 Defects particularly dislocations can provideanother vehicle to reduce the lattice thermal conduct-ivity in low-dimensional systems Clearly whether allor some of these strategies will work for the improve-ment of the energy conversion eYciency will dependon their impacts on the electronhole energy conver-sion capabilities More studies of these eVects shouldbe doneThe phonon size eVects in quantum wires and quan-tum dots are conceivably more signi cant than inthin lms and superlattices due to their increased

12 Comparison of measured thermal conductivity surfaceinterface area Up to now studies of theof highly dislocated Si Ge superlattice with thermal conductivity of quantum wires have beenpredictions of minimum thermal conductivity scarce There are a few experimental and theoreticaltheory for bulk Si and Ge and their multilayer

studies on the thermal conductivity of quantum dotcomposite latter is calculated based on Fourierarrays and nanostructured porous media1 0 7 1 0 8theory12 solid and dashed lines are fromTheoretically one can expect a larger thermal con-minimum thermal conductivity theories byductivity reduction in quantum wires compared toSlack101 and Cahill et al102 respectivelythin lms1 0 9 1 1 0 The measurements of the thermalconductivity in quantum wires have been challenginginterface eVects on the electron transport must also

be considered It may however be possible to capital- Recent measurements on the thermal conductivity ofindividual carbon nanotubes1 1 1 provide a possibleise on the diVerent wavelengths of electrons and

phonons such that phonons are scattered more approach for related measurements on thermoelectricnanowires of interest Nanowires for thermoelectricsdiVusely at the interface than electrons because the

dominant phonon wavelength is typically shorter however have very low k compared to carbon nano-tubes and this large diVerence in behaviour will createthan the electron wavelength

2 For transport perpendicular to the interfaces new challenges for measurements on nanowires ofinterest to thermoelectricityincreasing the phonon re ectivity is the key strategy

for reducing the thermal conductivity This could berealised by increasing the mismatch between the Bulk materialsproperties of the two constituents such as the density

The search for lsquophononndashglass electronndashcrystalrsquo bulkgroup velocity speci c heat and the phonon spectrummaterials has now moved beyond the traditionalbetween adjacent layers The eVects of interface rough-Bi

2Te

3based binary system Several classes of bulkness can be positive or negative depending on

materials have been discovered or re-investigated aswhether the diVuse phonon scattering actuallyinteresting for potential thermoelectric applicationsdecreases or increases the phonon re ectivity at theRepresentative among them are skutterudites clath-interfaces Experiments and modelling so far seem torates b-Zn

4Sb

3 half-Heusler intermetallic com-indicate that diVuse scattering is more eVective when

pounds and complex chalcogenides The phononthe mismatch in the material properties is not largerattler concept developed by Slack has proven to bePhonon con nement occurs due to the mismatchvery eVective in reducing the thermal conductivity ofbetween the bulk phonon dispersion relations and amaterials with open cages such as skutterudites andlarge diVerence in the dispersion favours more phononclathrates The highest Z T in lled skutteruditescon nement How much phonon con nement can bereached 1middot4 at ~600degC as shown in Fig 2 In thisachieved however is an open question Needless tosection several promising bulk thermoelectric mater-say however in pursuing the phonon thermal con-ials will be brie y reviewedductivity reduction the eVects of interfaces on the

electron transport must be considered Past studiesSkutteruditeson electron transport particularly in the development

of vertical-cavity surface-emitting lasers (VCSELs) Existence and compositionIn the early 1990s a systematic search for advancedindeed suggest the possibility of much stronger

reduction in the thermal conductivity than in the thermoelectric materials resulted in the lsquorediscoveryrsquoof a family of attractive semiconducting materialselectrical conductivity For example digitally or con-

tinuously graded interfaces are often used to reduce with the skutterudite crystal structure1 1 2 For state ofthe art thermoelectric materials such as PbTe andthe electrical resistivity of Bragg re ectors used for

VCSELs1 0 6 while it is likely that such graded Bi2Te

3alloys the number of isostructural compounds

is limited and the possibilities to optimise their bulkinterfaces can create more phonon back re ectionand thus a larger thermal conductivity reduction3 8 properties for maximum performance at diVerent

temperatures of operation are also very limited This3 Some long wavelength phonons may not lsquoseersquothe interfaces in structures such as superlattices The is not the case for the skutterudite family of materials

where 11 binary compounds and many solid solutionslocalisation of these phonons can further decrease thelattice thermal conductivity Using aperiodic rather and related phases are known to exist1 1 3 These



materials cover a large range of decomposition tem-peratures and band gaps which oVer the possibilityto adjust the composition and doping level for aspeci c temperature range of application An excellentin-depth review of skutterudites as novel thermoelec-tric materials was recently written by Uher1 1 4

The unit cell of the skutterudite structure (cubicspace group Im3 prototype CoAs

3) contains square

radicals [As4]4 ndash This anion which is located in the

centre of the smaller cube is surrounded by 8 trivalenttransition metal Co3 + cations The unit cell consistsof 8 of these smaller cubes or octants two of themempty and six of them containing the anions [As

4]4 ndash

in the centre This arrangement is necessary to main-tain the stoichiometric ratio Co3 + [As

4]4 ndash =4 3

Taking into account one-half of the full 32 atom unitcell and its empty octant one can represent theskutterudite formula as h T

4Pn

1 2 where h is the

13 Schematic of filled skutterudite 34 atom unitempty octant T is the transition metal and Pn is thecell of novel thermoelectric materials ndash eachpnicogen atom If considering a simple bondingcell contains 8 transition metal atoms Fe Ruscheme1 1 5 each transition metal contributes 9 elec-Os Co Rh Ir Ni Pd or Pt 24 pnicogen atomstrons and each pnicogen contributes 3 electrons toP As Sb (substitution by S Se or Te possible)

the covalent bonding for a valence electron count 2 rare earth atoms filling vacant octants in(VEC) total of 72 for each h T

4Pn

1 2unit The VEC skutterudite structure La Ce Pr Nd Sm Eu

is a useful number in determining the skutterudite Gd Th and Ucompositions that are semiconducting The valenceelectron count of 72 corresponds to a diamagnetic

It should be noted however that even if a simplesemiconductor for the skutterudite materials Uher1 1 4

VEC scheme may be used to explain the variousgave a detailed discussion on the bonding scheme

atomic composition permutations in skutteruditesthat favours the semiconducting behaviour There arethe actual picture is certainly quite a bit more compli-

eleven h T4Pn

1 2binary skutterudites reported in the

cated especially when trying to explain the transportliterature The nine semiconducting compositions are

properties of the more complex skutterudites in viewformed with all nine possible combinations of T=of some doping and compositional limitations Much

Co Rh Ir and Pn=P As Sb Two additionaleVort has been devoted in recent years to understand-

skutterudite phosphides were reported NiP3

anding the valence of transition metals and some rarePdP

3 However in these two compounds the total

earth lling atoms in ternary and lled skutteruditesVEC is 73 resulting in metallic behaviour1 1 6

but results are somewhat inconclusive at this point1 1 4The existence of many ternary skutterudites has

been determined Nine ternary compounds have been Transport propertiesThe transport properties of both n-type and p-typereported in the literature and at least 17 more have

been discovered1 1 7 Ternary skutterudite compos- conductive binary skutterudite compounds mostlyantimonides and arsenides have been thoroughlyitions are derived from binary compounds by keeping

a total VEC of 72 Using h Co4Sb

1 2(CoSb

3) as an characterised in the past few years (see eg Refs

124ndash126) Associated with a low hole eVective massexample substituting trivalent Co (Co3 + ) by divalentFe (Fe2 + ) and tetravalent Pd (Pd4 + ) results in very high hole mobilities low electrical resistivities

and moderate Seebeck coeYcients are obtained inh Fe2Pb

2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is

replaced by Sn and Te then h Co4Sn

6Te

6p-type skutterudites (Fig 14) Most of the samplesshown in this gure were not nominally doped They(CoSn

1 middot5Te

1 middot5) is obtained If substitutions occur on

both the transition metal and pnicogen sites then were grown using a gradient freeze technique fromnon-stoichiometric Sb rich melts1 2 7 One should noteh Fe

4Sb

8Te

4(FeSb

2Te) is obtained

A lled skutterudite structure is simply derived that the room temperature mobility values of p-typeskutterudites are very high (Fig 14a) about 10 tofrom the skutterudite structure by inserting one atom

in each empty octant as illustrated in Fig 13 A large 100 times higher than those for p-type Si and GaAsat similar carrier concentrations RhSb

3exhibits thenumber of these compounds have been known for

some time (see eg Refs 116 118ndash121) where the greatest hole mobility 8000 cm2 V ndash 1 s ndash 1 for a carrierconcentration of 2middot5 Ouml 101 8 cm ndash 3 which is about 70 lling atom is typically a rare earth lanthanoid

though other compositions with actinoids Th and times higher than p-type GaAs and still 5 times higherthan n-type GaAs For a comparable doping levelU1 2 1 1 2 2 as well as alkaline earths Ca Sr and Ba1 2 1 1 2 3

have also been reported For a typical lled skutterud- the carrier mobilities of n-type samples are about anorder of magnitude lower than the values achievedite composition such as LaFe

4P

1 2 the rare earth

element contributes 3 electrons but due to the on p-type samples However the much larger electroneVective masses and Seebeck coeYcients make n-typedivalent Fe (Fe2 + ) the total VEC is only 71 This

de cit results in metallic behaviour for most simple skutterudites promising candidates as wellUnfortunately the room temperature thermal con- lled ternary compounds Only CeFe

4P

1 2 UFe

4P

1 2and CeFe

4As

1 2have been reported as semiconductors ductivity of binary skutterudites (10ndash25 W m ndash 1 K ndash 1 )



order of magnitude lower in some ternary un lled or lled compositions Also as shown in the table highdoping levels in n-type CoSb

3result in a 70

reduction in lattice thermal conductivity an eVectcomparable to that obtained by forming solid solu-tions As a result Z T values as high as 0middot85 areobtained in the 850ndash900 K temperature range inheavily doped n-type CoSb

31 2 8 as shown in Fig 2

These experimental results thus demonstrate thatthe skutterudite family of compounds and alloys oVerextremely attractive possibilities for the search forhigh Z T materials Skutterudites have excellent elec-trical transport properties and the thermal conduct-ivities can be reduced by a factor of 20 down tonearly glassy characteristics Since the work ofChasmar and Stratton1 2 9 it has been obvious thathigh Z T values require materials with a high mobilityhigh eVective mass and low lattice thermal conduct-ivity where ideally one can separately optimise thepower factor and minimise the thermal conductivityor as summarised by Slack5 fabricate a PGEC Whilethe conceptual approach to high Z T has been knownfor over 40 years much eVort has been devoted into nding out if skutterudites could truly be such PGECmaterials that is conserving the excellent electricalproperties of the binary compounds while reducingthe thermal conductivity to values substantially lowerthan that of state of the art of Bi

2Te

3alloys

However a number of experimental and theoreticalresults published in recent years on both ternaryun lled and lled skutterudite compounds indicatethat their electronic band structure and transportproperties are vastly diVerent from the pure binarycompounds1 3 0 1 3 1 For example the study of suchternary compounds as FeSb

2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3

derived from high carrier mobilityCoSb

3and RhSb

3shows that they are heavily doped

semiconductors with carrier concentration valuesranging from 1 Ouml 102 0 to 1 Ouml 102 1 cm ndash 3 Comparingthese compounds to CoSb

3at such high carrier

concentrations they are characterised by signi cantlylower carrier mobility and lattice thermal conduct-

14 a Hall mobility variation as function of carrier ivity values as much as 50 to 75 lower Such lowconcentration and b Seebeck coefficient as lattice thermal conductivity values are somewhatfunction of temperature for binary skutteruditecompounds samples were not nominally

Table 1 Room temperature conductivity typedoped127

carrier concentration N and latticethermal conductivity kp of variousskutterudite compounds and alloysis too high to result in high Z T values Substantialincluding filled compositionsreductions in the lattice thermal conductivity must

be obtained to achieve values comparable to thoseCompound Conductivity N cm ndash 3 kp W m ndash 1 K ndash 1

of state of the art thermoelectric materialsCoP3 p 3 Ouml 1019 23middot7(1ndash4 W m ndash 1 K ndash 1 ) Several approaches to theCoAs3 p 4 Ouml 1018 14middot0reduction of the lattice thermal conductivity in skut-CoSb3 p 9 Ouml 1018 10middot3terudites have been pursued heavy doping and for-CoSb3 n 1 Ouml 1021 3middot4mation of solid solutions and alloys as well as theCoP1middot5As1middot5 p 5 Ouml 1018 3middot8

study of novel ternary and lled skutterudite com- Co0middot88 Ir0middot12Sb3 p 1 Ouml 1019 2middot9pounds All those approaches have resulted in skut- FeSb2 Te p 5 Ouml 1020 2middot3terudite compositions with substantially lower OsSb2 Te p 4 Ouml 1018 2middot1thermal conductivity values The room temperature RuGe0middot2 Sb2 Te0middot8 p 9 Ouml 1019 1middot4

Ru0middot5 Pd0middot5 Sb3 p 1 Ouml 1020 1middot5lattice thermal conductivity and carrier concentrationIrSn1middot5 Se1middot5 p 3 Ouml 1019 2middot7values of selected skutterudite compounds and alloysCeRu4P12 p 1 Ouml 1020 8middot0are reported in Table 1 for comparison It is interes-CeFe4Sb12 p 3 Ouml 1021 1middot6ting to note that compared to binary compoundsNd0middot7 Ru2 Co2Sb12 p 2 Ouml 1020 1middot8the lattice thermal conductivity values can be one



surprising considering that the atomic mass andvolume diVerences introduced by the substitutinganioncation are fairly small Fe and Ni for Co Ruand Pd for Rh Te for Sb A possible explanationgiven for the unusually high phonon scattering rateis that transition metal elements have mixed valencestates and electrons are transferred between thediVerent ions thus scattering the phonons in thisprocess1 3 2 1 3 3 This would be consistent with lowcarrier mobilities as well as with the experimentaldiYculties encountered in controlling their electricalproperties since changes in carrier concentration arenot easy to achieve because dopants can be compen-sated by small uctuations in the overall valence ofthe transition metals

Interestingly enough the search for high Z T valuesin skutterudites has been most successful so far incompounds that do not share the exceptional carriermobilities of the pure binary compounds but ratherhave a combination of high degeneracy unusually 15 Room temperature lattice thermal con-

ductivity as function of filling atom rattlinglarge Seebeck coeYcient values and low lattice ther-amplitude for various fully filled and fully densemal conductivities lled and partially lled skutterud-filled skutterudite compounds carrier con-ites A maximum Z T value of 1middot4 has been achievedcentrations are also reported to account forto date at a temperature of 875 K in Ce

fFe

4 ndash xCo

xSb

1 2 additional charge carrier scattering of phononsand LafFe

4 ndash xCo

xSb

1 2 where flt1 and 0ltx lt

41 3 4 1 3 5

The initial appeal of introducing a lling atom into itions has proven quite challenging from a materialssynthesis standpoint Electron microprobe analysis ofthe skutterudite structure was to substantially reduce

the lattice thermal conductivity of the original binary a series of CefFe

4 ndash xCo

xSb

1 2compounds has demon-

strated that the fraction of Ce lling f decreases withcompound with minimal decrease in the carriermobility and thus to achieve a lsquoPGECrsquo The heavy increasing substitution of Fe by Co In addition to

Co substitution of Fe by Ni and Ru and the substi- lling atom would lsquorattlersquo within its octant lsquocagersquo andthus scatter phonons quite eVectively Also because it tution of Ru by Pd have also been investigated

recently The variations of the lling fraction f as ais lling an empty octant its contribution to theelectrical transport would be minimal though the function of x are plotted in Fig 16 for the four

diVerent ranges of compositions The two solid linesincreased phonon scattering rate should somewhatimpact the carrier scattering rate (carrierndashphonon represent the expected transition for Ni and Co from

p-type to n-type (when the VEC reaches 72) takinginteraction) However it became rapidly clear that lled skutterudites are quite diVerent from their into account both f and x variations When Fe is

totally replaced by Co only a very small amountbinary relatives and that their low lattice thermalconductivities result from a combination of eVects of Ce remains in the sample ( f=0middot07) The chain

dotted line was calculated based on a CeFe4Sb

1 2ndashoften quite diYcult to separate from each other

rattling carrierndashphonon scattering (high doping Ce0 middot0 6 5

Co4Sb

1 2range of lsquosolid solutionrsquo compo-

sitions CefFe

4 ndash xNi

xSb

1 2compositions with x gt1middot5levels) mixed valence of the transition metals point

defect scattering have not yet been synthesised but it is clear thatat equivalent concentrations Ni substitution res-If comparing fully lled compositions and taking

into account charge carrier concentration levels how- ults in less Ce lling than Co substitution Howeverbecause Ni donates two electrons instead of only oneever it can be shown that the lattice thermal conduct-

ivity does indeed decrease signi cantly with increasing for Co when replacing Fe the decrease in carrierconcentration and the corresponding change in prop-rattling amplitude of the lling atom as shown in

Fig 15 The rattling amplitude is de ned as the erties with increasing x is much stronger forCe

fFe

4 ndash xNi

xSb

1 2 One can see from these results thatdiVerence between the void ller covalent radius and

the radius of the cage1 3 6 The rattling amplitude semiconducting compositions can be obtained for Corich compositions However they typically showincreases are the largest when going from the phos-

phides to the antimonides which have larger unit cells mixed conduction eVects at room temperature1 1 7

Recent theoretical studies on n-type lled skutteru-and cage sizes and very little when going from Febased to Ru based compositions dites1 3 7 and new experimental materials preparation

approaches through high pressure high temperatureThe most recent research eVorts on skutteruditeshave focused on improving Z T over a wide temper- synthesis or the use of thin interdiVusing layers1 3 8 1 3 9

may eventually allow for the characterisation ofature range by attempting to conserve the excellentsemiconducting behaviour of the un lled binary skut- lsquooptimisedrsquo skutterudite compositions with perhaps

high Z T values at room temperature In the mean-terudites when introducing a lsquocompensatingrsquo atom forthe addition of the lsquo llingrsquo atom into the structure time skutterudites are now being actively pursued

for introduction into advanced thermoelectric powerand going back to a VEC of 72 However preparinglightly doped extrinsic p-type and n-type compos- generation devices and systems with the potential



frequency phonons by the Sr atoms that are looselybonded inside the structure and are free to rattlearound their atomic positions A good indicator ofthe possible motion of the guest atoms is the atomicdisplacement parameter (ADP) that is a measure ofthe mean-square displacement amplitude of an atomaround its equilibrium site Unusually large ADPvalues have been measured for guest atoms such asCs and Sr in Ge based clathrates1 4 3 Sales et al haverecently proposed a model to estimate the latticethermal conductivity based on ADP values1 4 5 Atroom temperature the thermal conductivity ofSr

8Ga

1 6Ge

3 0is close to that of amorphous Ge Low

thermal conductivity values have also been observedin other Ge based clathrates Silicon based clathratestend to have higher thermal conductivity values Agood thermoelectric material must combine a lowlattice thermal conductivity with good electronicproperties The electronic properties of several Si Ge

solid lines represent expected transition for Ni and Coand Sn based clathrates have been measured Metallicfrom p-type to n-type (when VEC reaches 72) taking intoto semiconducting behaviour can be achieved byaccount both f and x variations when Fe is totally replaced

by Co only very small amount of Ce remains in sample varying the doping andor composition Both n- and(f=0middot07) chain dotted line was calculated based on p-type materials can be obtained by the same processCeFe4Sb12ndashCe0middot065 Co4 Sb12 range of lsquosolid solutionrsquo

Good Seebeck coeYcient values up to shy 300 mV K ndash 1compositions (CefFe4 ndash x Nix Sb12 compositions with xgt1middot5

have been achieved To date the best dimensionlesshave not yet been synthesised but it is clear that atequivalent concentrations Ni substitution results in less gure of merit Z T obtained for n-type clathrateCe filling than Co substitution) compounds is about 0middot34 with a projected value gt1

16 Ce filling fraction f for CefFe4 ndash xMx Sb12 samples above 700 K The ability of engineering clathrateas function of x Fe substitution by M with compounds with glass-like thermal conductivity com-M=Co Ni and Ru bined with their relatively good electronic properties

has ranked clathrates among PGEC materialsalthough full decoupling between their electronic andfor 15 conversion eYciency in the 25ndash700degC tem-thermal properties remains to be demonstrated inperature range1 4 0these materials Nevertheless the encouraging resultsobtained to date combined with the numerous options

Other novel bulk materials for optimisation warrant further investigations ofClathrates these interesting materialsAmong other materials of interest as potential new

b-Zn4Sb3thermoelectric materials are clathrates A clathrateAnother material re-investigated recently for thermo-material can be de ned as one with a lattice thatelectric power generation is b-Zn

4Sb

3with a hexag-contains voids that can be lled with guest atoms or

onal rhombohedric crystal structure and space groupmolecules Silicon Ge and Sn based clathrates lledR3C as shown in Fig 17 Caillat et al1 4 6 havewith alkaline atoms have been reported1 4 1 1 4 2 andmeasured the thermal conductivity of polycrystallinehave recently been investigated for their thermoelec-b-Zn

4Sb

3samples The lattice thermal conductivitytric properties1 4 3 They can be divided into two major

is nearly temperature independent between 300 andgroups type I and type II Both types have a cubic650 K with a room temperature lattice thermal con-unit cell but diVer according to the number and sizeductivity of 0middot65 W m ndash 1 K ndash 1 at 300 K nearly 2 timesof the voids present in the structure Ternary com-lower than that of Bi

2Te

3alloys This remarkably lowpounds have also been reported There are numerous

value and unusual temperature dependence (a 1Tpossible compositional structural and lling vari-temperature dependence is usually observed) can beations in those materials resulting in vastly diVerentattributed to the relatively complex crystal structureelectronic properties ranging from semimetallic toas well as the presence of vacancies in the lattice Thesemiconducting Recently Nolas et al1 4 4 have meas-electronic transport properties are typical of a semi-ured the thermal conductivity of semiconductingmetal with low electrical resistivity that increases withSr

8Ga

1 6Ge

3 0polycrystalline samples and observed a

increasing temperature The Seebeck coeYcient alsotemperature dependence typical of amorphous mater-increases with increasing temperature and peaks atial These results have triggered extensive theoretical675 K with a value of about 200 mV K ndash 1 This rela-and experimental research eVorts to synthesise severaltively high Seebeck value for a metal is the result ofof these materials and further understand their pecul-a fairly large eVective mass1 4 6 The best Z T obtainediar transport properties A comprehensive review ofto date on polycrystalline samples is about 1middot4 atthese materials has recently appeared in the675 K (Fig 2)1 4 6 Above 765 K b-Zn

4Sb

3transformsliterature1 4 3

into c-Zn4Sb

3that has poorer thermoelectric prop-The low temperature T 2 temperature dependence

erties Band structure calculations1 4 7 predict a metallicobserved by Nolas et al1 4 4 for Sr8Ga

1 6Ge

3 0polycrys-

talline samples was attributed to the scattering of low behaviour with improved thermoelectric performance



coeYcient values Large power factors on the orderof 25ndash30 mW cm ndash 1 K ndash 2 have since been experimen-tally obtained for several of these materials egZrNiSn and HfNiSn The impact on the electronicproperties of compositional variations andor atomicsubstitutions on the various sublattices has beeninvestigated showing that both doping level andconductivity type can be altered While the powerfactors are promising the thermal conductivity ofternary compounds such as ZrNiSn and HfNiSn israther high The total thermal conductivity (which isessentially the lattice thermal conductivity for thesematerials) ranges from 5middot9 to 17 W m ndash 1 K ndash 1 forZrNiSn1 5 1 The spread in the values was primarilydue to the diVerence of structural quality introducedfor example by annealing the samples Attempts toreduce the lattice thermal conductivity of these mater-ials using mass-defect scattering on the various sub-lattices were made Although lattice thermalconductivity values of 5ndash6 W m ndash 1 K ndash 1 were obtainedfor alloys these values are still too high compared tothose obtained for state of the art thermoelectricalloys such as Bi

2Te

3alloys EVorts should therefore17 Schematic representation of Zn4Sb3 crystal

focus on further reducing the lattice thermal conduct-structure illustrating various Sb and Znivity of these materials that otherwise possess impress-atomic sitesive electronic properties that can be tuned throughdoping and alloying Possible schemes for reducingfor lower doping levels Little success has howeverthe thermal conductivity may however be somewhatbeen obtained experimentally to optimise the dopinglimited apart from the introduction of point defectlevel of this compound b-Zn

4Sb

3forms a full range

scattering since these materials are not lsquocage-likersquoof solid solutions with the isostructural compoundmaterials such as skutterudites or clathrates MostCd

4Sb

3 Low temperature thermal conductivity

transport property measurements on half-Heuslermeasurements on Zn4 ndash x

CdxSb

3mixed crystals

alloys to date have been limited to low temperaturesshowed a nearly temperature independent variationbut these materials may actually be quite interestingof the thermal conductivity which is lower than forat temperatures up to 800ndash900 K considering thatb-Zn

4Sb

3(Ref 148) mostly due to point defect scat-

they typically behave as semimetals at temperaturestering Zn4 ndash x

CdxSb

3mixed crystals appear to be less

below 300 K and semiconductors above that temper-stable than b-Zn4Sb

3itself and a Z T maximum of

ature Transport property measurements above 300 K1middot4 at about 400 K was obtained for these mixedwould therefore be of interest to fully assess thecrystals EVorts to incorporate b-Zn

4Sb

3into

potential of these materials for thermoelectricadvanced high eYciency thermoelectric unicouplesapplicationsare in progress1 4 9 Further optimisation of the prop-

erties of these compounds is limited because of the Complex chalcogenidesdiYculties to dope them and the restricted compos- While several materials with Z T gt1 have been ident-itional variations possible i ed above room temperature there is a great need

for new thermoelectric materials with Z T gt1 forHalf-Heusler intermetallic compoundscooling applications At and below room temperatureHalf-Heusler intermetallic compounds with the gen-only two materials have been known for many yearseral formula MNiSn (M=Zr Hf Ti) have alsoto have decent thermoelectric properties BindashSb andattracted considerable interest as potential new ther-Bi

2Te

3alloys Past studies suggest new low temper-moelectric materials1 5 0 These materials possess the

ature semiconducting thermoelectric materials areMgAgAs structure and are closely related to the fulllikely to be found in narrow band gap materials1 4Heusler compounds MNi

2Sn which are metals

An extensive eVort at Michigan State UniversityReplacing one Ni atom by an ordered lattice ofhas focused on a number of new chalcogenides com-vacancies leads to a gap formation in the density ofposed mostly of heavy elements As a result a numberstates and to a semiconducting character with bandof potential new materials for low temperature ther-gap values on the order of 0middot1 to 0middot2 eV As a resultmoelectric applications have been identi ed A com-of the large electron eVective mass in these materialsprehensive review of these materials has recentlyhigh Seebeck coeYcients are typically obtained atappeared in the literature1 5 3 A number of complex300 K and higher EVective masses of 2ndash3 m

0were

phases have been prepared preferably by a ux tech-estimated by Uher et al1 5 1 for ZrNiSn and Seebecknique and several new compounds have been ident-coeYcients as high as shy 300 mV K ndash 1 were measuredi ed Among the materials investigated are theat 300 K for this material It was originally suggestedsulphides KBi

6 middot3 3S

1 0and K

2Bi

8S

1 3 the selenides b-by Cook et al1 5 2 that these materials might be good

K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe

1 5thermoelectric materials considering their combi-nation of low electrical resistivity with large Seebeck (A=K Rb) compounds and the tellurides ABiTe



and APbBiTe A common feature for most of these Thermoelectric materials require high doping withcarrier concentrations of ~101 9 cm ndash 3 Carefulmaterials is their low thermal conductivity compar-

able or even lower than that of Bi2Te

3alloys Many optimisation of the doping concentration is necessary

Thus an experimental thermoelectric research pro-of these compounds show very anisotropic transportproperties Perhaps the most promising compound gramme must be prepared in the synthesis and

optimisation as well as fast characterisation turn-identi ed to date is CsBi4Te

6 This compound has a

layered anisotropic structure with Cs ions between around capabilityMost thin lms in electronic devices are on the[Bi

4Te

6] layers The ADPs of the Cs ions are 1middot6

times greater than those of the Bi and Te atoms order of submicrometres in the lm thicknessThermoelectric thin lms should typically be thickerCrystals of CsBi

4Te

6grow with a needlelike morph-

ology and are stable in air and water The crystals both for characterisation purposes and for deviceapplications For thermoelectric devices in the cross-are amenable to doping and SbI

3 BiI

3 and In

2Te

3have been successfully used to optimise the carrier plane direction lms thicker than a few micrometres

or even much thicker are desirable in terms of sustain-concentration of this material The power factor canbe maximised through doping and a maximum power ing a reasonable temperature diVerential For thermo-

electric devices intended for use for in-plane transportfactor value of about 50 mW cm ndash 1 K ndash 2 was obtainedat 185 K for the p-type material1 5 3 The total thermal applications the reverse heat ow from the supporting

substrate must be minimised This means either theconductivity along the growth axis is about1middot5 W m ndash 1 K ndash 1 at 300 K and is essentially constant removal of the substrate or depositing very thick

lms Thus both in-plane and cross-plane devicesdown to 100 K This atypical temperature dependencesuggests again that the rattling Cs ions signi cantly demand relatively thick lms while quantum or

classical size eVects typically require individual layerscontribute to phonon scattering in this compoundThe best Z T to date obtained along the needle of the order of several tens of angstroms Such con-

icting requirements impose a severe limit for thedirection is 0middot82 for the p-type material (Fig 2)slightly better than p-type Bi

2Te

3at this temperature practical scale up of materials synthesis methods

although in research many thin lm depositionThe In2Te

3lsquodopedrsquo n-type material has poorer ther-

moelectric properties Nevertheless p-type CsBi4Te

6methods such as molecular beam epitaxy metallo-organic chemical vapour deposition pulsed laseris the rst compound identi ed in the low temperature

range to match or even outperform Bi2Te

3alloys deposition and sputtering are all being explored

Thermoelectric materials employ relatively largeWhether or not this compound can be furtheroptimised through doping andor alloying will need numbers of alloy compositions For example

Si0 middot8

Ge0 middot2

is typically used in bulk thermoelectricto be determined in the future as well as its mechanicalstability under thermal stresses to warrant its practical generators For comparison electronic devices based

on SiGe alloys such as heterojunction bipolar transis-use in devicesPentatelluride materials such as HfTe

5and ZrTe

5tors use 1ndash5Ge concentration A larger concen-tration of Ge is required for thermoelectricsand their alloys have been considered promising new

thermoelectric materials at low temperatures because applications to create more thermal conductivityreduction Even in superlattices a relatively largeof their relatively large Seebeck coeYcient values at

low temperatures which combined with relatively low equivalent Ge concentration is needed for suYcientthermal conductivity reduction For superlatticeselectrical resistivity values result in large power factor

values1 5 4 Their electronic properties can be tuned made of materials with a large mismatch in theirlattice constants buVer layers are needed To growthrough alloying and doping but the challenge for

these materials is to reduce their lattice thermal 2020 A SiGe superlattices for example gradedbuVer layers of SiGe alloys from 1 to 5 mm withconductivity Another challenge lies in the very aniso-

tropic nature of these materials that requires the continuously varying Ge concentrations have beenused These buVer layers complicate the characteris-growth and characterisation of single crystals for

transport properties Single crystal whiskers were ation and usually degrade the device performanceThe characterisation of thermoelectric propertiesobtained by a vapour transport technique but

measurements on these small crystals oVer great of potentially interesting thermoelectric materialsparticularly samples consisting of low-dimensionalchallenges Further investigations will be required to

determine whether or not the transport properties of structures imposes an even greater challenge Evenfor bulk materials thermal conductivity measure-these materials can be further optimised and if thermal

conductivity values close to those for state of the art ments are never easy and are prone to large uncer-tainty For low-dimensional structures thermalthermoelectric materials can be obtained without

signi cantly degrading their electronic properties conductivity measurements are much trickier Themost widely used method for cross-plane thermalconductivity measurements is the 3v method thatSpecial challenges in materialsrelies on the deposition of small heaters that also act

synthesis and characterisation as temperature sensors8 7 The principle of the 3vmethod is simple but when applied to speci c low-Compared to semiconductor materials for electronic

and optoelectronic devices thermoelectric materials dimensional structures it can be quite involved dueto the following factorsimpose diVerent sets of materials requirements which

in turn create new challenges in their materials syn- 1 Thermoelectric lms are conducting thus aninsulator is needed for electrical isolation between thethesis and characterisation Some of these special

challenges will be brie y discussed here sensor and the sample The insulator has unknown



thermal conductivity Often a diVerential method electric transport in both low-dimensional and bulkmaterials However there is much left to be done inis used1 5 5

2 Superlattices are grown on substrates and new materials syntheses characterisation physicalunderstanding and device fabrication It is hopedbuVers whose properties are not exactly known

Although the principle of the 3v method allows the that this review will arouse broader interest in thermo-electrics research from the materials research com-measurements to be made on the thermal conductivity

of the substrate the determination of a high thermal munity Meanwhile the authors would like toemphasise that thermoelectric materials research is aconductivity substrate is actually subject to quite

large uncertainties The buVer layers typically have multidisciplinary endeavour and requires close collab-oration between researchers in diVerent elds tounknown and anisotropic thermal conductivities that

cannot be easily determined address issues in materials theory characterisationand eventually devices3 Superlattices also have anisotropic properties

and thus care must be taken to ascertain whichdirection is being measured Through careful model- Acknowledgementsling both the in-plane and cross-plane direction ther-

Two of the authors (GC and MSD) gratefullymal conductivity can be determined4 0 Several otheracknowledge their collaborators in thermoelectricfactors such as the thermal property contrast betweenresearch including Professors R Gronsky J-P Issithe lm and the substrate and the lm heat capacityT D Sands K L Wang Dr J Heremans andeVects are discussed in detail by Borca-TasciucT Harman and contributions from all students inet al1 5 5 In addition to the 3v method other methodstheir respective research groups The authors arethat have been used often are the ac calorimetrygrateful for support for this work by DoD MURImethod for determining the thermal conductivityN00014ndash97ndash1ndash0516 (GC and MSD) US Navyalong the lm plane direction1 5 6 and the pump-and-Contract N00167ndash98ndashKndash0024 (MSD) DARPAprobe method for determining the cross-plane thermalContract N66001ndash00ndash1ndash8603 (MSD) DARPAconductivity8 0HERETIC Project (J-PF and GC) JPL 004736ndash001The determination of the Seebeck eVect is usually not(J-PF and GC) and NSF Grant DMR 01ndash16042considered to be a big challenge for bulk materials(MSD)but it can be quite tricky for superlattices that are

grown on substrates and buVers because the substratemay make a large contribution to the overall meas- Referencesured properties Although corrections can be made

1 h j goldsmid lsquoThermoelectric refrigerationrsquo 1964 Newbased on simple circuit theory there is no strong York Plenum Pressevidence that such corrections are reliable Several 2 a f ioffe lsquoSemiconductor thermoelements and thermoelectric

coolingrsquo 1957 London Infosearch Ltdmethods have been attempted in the past for measur-3 g n hatsopoulos and j kaye J Appl Phys 1958 29ing the in-plane Seebeck coeYcient and conductivity

1124ndash1126of superlattices by removing the substrate or by4 g d mahan J Appl Phys 1994 76 4362ndash4366

growing the samples on insulators2 8 3 6 1 5 7 1 5 8 Even 5 g slack in lsquoCRC handbook of thermoelectricsrsquo (ed D MRowe) 407ndash440 1995 Boca Raton CRC Pressfor samples grown on insulators such as SiGe super-

6 l d hicks and m s dresselhaus Phys Rev B 1993 47lattices grown on silicon-on-insulator substrates the12727ndash12731additional contributions of the buVer and of the

7 t c harman p j taylor d l spears and m p walshsilicon layer must be taken into account In the cross- J Electron Mater 2000 29 L1ndashL4plane direction direct measurements of the Seebeck 8 b y moyzhes and v nemchinsky Proc Int Conf on

lsquoThermoelectricsrsquo ICTrsquo92 1992 232ndash235 (Appl Phys LettcoeYcient have only been reported very recently1 5 91998 73 1895ndash1897)One can appreciate the diYculties involved by exam-

9 a shakouri and j e bowers Appl Phys Lett 1997 71 1234ining equation (1) which requires measurements of10 g d mahan Semicond Semimet 2001 71 157ndash174

both the temperature diVerence and the voltage drop 11 r venkatasubramanian Semicond Semimet 2001 71175ndash201across two equivalent points In the cross-plane direc-

12 g chen Semicond Semimet 2001 71 203ndash259tion this measurement becomes extremely challeng-13 t m tritt (ed) in lsquoRecent trends in thermoelectric materialsing since the thickness of the lm is typically on the

researchrsquo Semicond Semimet 2001 69ndash71order of 1ndash10 mm The Z T values in the cross-plane 14 g d mahan Solid State Phys 1998 51 81ndash157direction for Bi

2Te

3Sb

2Te

3superlattices1 1 have been 15 g s nolas d t morelli and t m tritt Ann Rev Mater

Sci 1999 29 89ndash116measured using the transient Harman method116 f j disalvo Science 1999 285 (5428) 703ndash706It should be emphasised that all the measurements17 g s nolas and g a slack Am Sci 2001 89 (2) 136ndash141should be done in the same direction and preferably18 Proc Int Conf on lsquoThermoelectricsrsquo eg ICTrsquo97 (IEEE Cat

also on the same sample For low-dimensional struc- no 97TH8291) ICTrsquo98 (IEEE Cat no 98TH8365) ICTrsquo99(IEEE Cat no 99TH8407) ICTrsquo00 (Babrow Press Walestures this has proven to be a very diYcult goal toUK ISBN 0951928627) ICTrsquo01 (IEEE Cat no 01TH8589)achieve For these reasons the Z T values of low-

19 MRS Proceedings on Thermoelectrics 478 (1997) 545 (1998)dimensional structures plotted Fig 2 should be626 (2000) Warrendale PA Materials Research Society

further con rmed 20 n w aschroft and n d mermin lsquoSolid state physicsrsquo 1976Forth Worth Saunders College Publishing

21 m s dresselhaus y-m lin s b cronin o rabin m rConcluding remarks black g dresselhaus and t koga Semicond Semimet

2001 71 1ndash121In summary the recent resurgence in thermoelectric22 j goldsmid in Proc 17th Int Conf on lsquoThermoelectricsrsquo

materials research has led to quite a large increase in ICTrsquo98 1998 25ndash28 (IEEE Cat no 98TH8365)23 d j singh Semicond Semimet 2001 70 125ndash178Z T values and signi cant new insights into thermo-



24 l d hicks and m s dresselhaus Phys Rev B 1993 47 61 a n korotkov and k k likharev Appl Phys Lett 199916631ndash16634 75 2491ndash2493

25 l d hicks lsquoThe eVect of quantum-well superlattices on the 62 t zeng and g chen Proc Int Mech Eng Congress andthermo-electric gure of meritrsquo PhD thesis MIT June 1996 Exhibition ASME HTDndashVol 366ndash2 2000 361ndash372

26 t koga t c harman x sun and m s dresselhaus Phys 63 t zeng and g chen J Appl Phys 2002 92 3152Rev B 1999 60 14286ndash14293 64 g chen and t zeng Microscale T hermophys Eng 2001

27 l d hicks t c harman and m s dresselhaus Phy Rev B 5 71ndash881996 53 10493ndash10496 65 a shakouri c labounty j piprek p abraham and j e

28 x sun s b cronin j l liu k l wang t koga m s bowers Appl Phys Lett 1999 74 88ndash89dresselhaus and g chen Proc 18th Int Conf on 66 g h zeng a shakouri c la bounty g robinson e crokelsquoThermoelectricsrsquo ICTrsquo99 1999 652ndash655 (IEEE Cat no p abraham x f fan h reese and j e bowers Electronics99TH8407) Lett 1999 35 2146ndash2147

29 t koga s b cronin m s dresselhaus j l liu and k l 67 x f fan g h zeng c labounty j e bowers e crokewang Appl Phys Lett 1999 77 1490ndash1492 c c ahn s huxtable a majumdar and a shakouri Appl

30 t c harman d l spears d r calawa s h groves and Phys Lett 2001 78 1580ndash1582m p walsh Proc 16th Int Conf on lsquoThermoelectricsrsquo 68 x f fan g zeng e croke c labounty c c ahnICTrsquo97 1997 416ndash423 (IEEE Cat no 97TH829) d vashaee a shakouri and j e bowers Electron Lett

31 t c harman d l spears and m p walsh J Electron 2001 37 126ndash127Mater Lett 1999 28 L1ndashL4 69 c labounty a shakouri p abraham and j e bowers

32 j o sofo and g d mahan Appl Phys Lett 1994 65 Opt Eng 2000 39 2847ndash28522690ndash2692 70 c labounty a shakouri and j e bowers J Appl Phys

33 d a broido and t l reinecke Phys Rev B 1995 51 2001 89 4059ndash406413797ndash13800 71 r tsu and r f greene Electrochem Solid State Lett 1999

34 t koga lsquoConcept and applications of carrier pocket engineer- 2 645ndash647ing to design useful thermoelectric materials using superlattice 72 h b g casimir Physica 1938 5 495ndash500structuresrsquo PhD thesis Harvard University June 2000 73 c l tien and g chen in Proc Am Soc Mech Eng ASME

35 t koga x sun s b cronin and m s dresselhaus Appl HTDndashVol 227 1992 1ndash12 (J Heat T ransf 1994 116Phys Lett 1998 73 2950ndash2952 799ndash807)

36 t koga x sun s b cronin and m s dresselhaus Appl 74 g chen Ann Rev Heat T ransf 1996 7 1ndash57Phys Lett 1999 75 2438ndash2440 75 r venkatasubramanian Naval Res Rev 1996 58 44ndash54

37 t yao Appl Phys Lett 1987 51 1798ndash1800 76 r venkatasubramanian e silvola and t colpitts Nature38 g chen c l tien x wu and j s smith J Heat T ransf 2001 413 597ndash602

1994 116 325ndash331 77 i yamasaki r yamanaka m mikami h sonobe y mori39 x y yu g chen a verma and j s smith Appl Phys Lett and t sasaki Proc 17th Int Conf on lsquoThermoelectricsrsquo

1995 67 3554ndash3556 1996 68 1303 ICTrsquo98 1998 210ndash213 (IEEE Cat no 98TH8365)40 w l liu t borca-tasciuc g chen j l liu and k l 78 m n touzelbaev p zhou r venkatasubramanian and

wang J Nanosci Nanotechnol 2001 1 39ndash42 k e goodson J Appl Phys 2001 90 763ndash76741 h beyer j nurnus h bottner l lambrecht t roch and 79 w s capsinski and h j maris Physica B 1996 220 699ndash701

g bauer Appl Phys Lett 2002 80 1216ndash1218 80 w s capinski h j maris t ruf m cardona k ploog42 t c harman p j taylor m p walsh and b e laforge and d s katzer Phys Rev B 1999 59 8105ndash8113

Science 2002 297 222981 s m lee d g cahill and r venkatasubramanian Appl

43 x sun z zhang and m s dresselhaus Appl Phys Lett Phys Lett 1997 70 2957ndash29591999 74 4005ndash4007

82 t borca-tasciuc w l liu t zeng d w song c d moore44 z b zhang lsquoFabrication characterization and transport

g chen k l wang m s goorsky t radetic r gronskyproperties of bismuth nanowire systemsrsquo PhD thesis MIT

t koga and m s dresselhaus Superlattices M icrostructFebruary 1999

2000 28 119ndash20645 y-m lin lsquoFabrication characterization and theoretical mode-

83 r venkatasubramanian Phys Rev B 2000 61 3091ndash3097ling of Te-doped Bi nanowire systems for thermoelectric

84 t borca-tasciuc d achimov w l liu g chen h-w renapplicationsrsquo MS thesis MIT January 2000

c-h lin and s s pei Microscale T hermophys Eng 200146 z zhang x sun m s dresselhaus j y ying and h herem-

5 225ndash231ans Appl Phys Lett 1998 73 1589ndash1591

85 s t huxtable a shakouri c labounty x fan p abraham47 j heremans c m trush z zhang x sun m s dresselhaus

y j chiu j e bowers and a majumdar Microscalej y ying and d t morelli Phys Rev B 1998 58T hermophys Eng 2000 4 197ndash203R10091ndashR10095

86 d w song w l liu t zeng t borca-tasciuc g chen48 z zhang x sun m s dresselhaus j y ying and j herem-c caylor and t d sands Appl Phys Lett 2000 77ans Phys Rev B 2000 61 4850ndash48613854ndash385649 j heremans c m trush y-m lin s cronin z zhang m s

87 s m lee and d g cahill J Appl Phys 1997 81 2590ndash2595dresselhaus and j f mansfield Phys Rev B 2000 6188 p hyldgaard and g d mahan Phys Rev B 1997 562921ndash2930

10754ndash1075750 y m lin x sun and m s dresselhaus Phys Rev B 200089 s tamura y tanaka and h j maris Phys Rev B 1999 6062 4610ndash4623

2627ndash263051 b lenoir m cassart j p michenaud h sherrer and90 w e bies r j radtke and h ehrenreich J Appl Physs sherrer J Phys Chem Solids 1996 57 88ndash89

2000 88 1498ndash150352 o rabin y m lin and m s dresselhaus Appl Phys Lett91 a a kiselev k w kim and m a stroscio Phys Rev B2001 79 81ndash83

2000 62 6896ndash689953 y-m lin s b cronin o rabin j y ying and m s92 b yang and g chen Microscale T hermophys Eng 2001dresselhaus Appl Phys Lett 2001 79 677ndash679

5 107ndash11654 l w whitlow and t hirano J Appl Phys 1995 7893 a balandin and k l wang J App Phys 1998 845460ndash5466

6149ndash615355 r j radtke h ehrenreich and c h grein J Appl Phys94 g chen J Heat T ransf 1997 119 220ndash229 (Proc 1996 Nat1999 86 3195ndash3198

Heat Transfer Conf ASME HTDndashVol 323 1996 121ndash130)56 g d mahan J App Phys 1994 76 4362ndash436695 p hyldgaard and g d mahan in lsquoThermal conductivityrsquo57 g d mahan and l m woods Phys Rev Lett 1998 80

Vol 23 172ndash181 1996 Lancaster Technomic4016ndash401996 g chen and m neagu Appl Phys Lett 1997 71 2761ndash276358 g d mahan j o sofo and m bartkowiak J Appl Phys97 g chen Phys Rev B 1998 57 14958ndash14973 (see also Proc1998 83 4683ndash4689

1996 Int Mech Eng Congr 1996 DSCndashVol 59 13ndash24)59 c b vining and g d mahan J Appl Phys 1999 8698 g chen J Heat T ransf 1999 121 945ndash9536852ndash685399 k e goodson and y s ju Ann Rev Mater 1999 29 261ndash29360 n m miskovsky and p h cutler Appl Phys Lett 1999 75

2147ndash2149 100 k e goodson Ann Rev Heat T ransf 1995 6 323ndash353



101 g a slack Solid State Phys 1979 34 1ndash71 131 j o sofo and g d mahan MRS Symp Proc 545 315 1999Warrendale PA Materials Research Society102 d g cahill s k watson and r o pohl Phys Rev B

132 g a slack j-p fleurial and t caillat Naval Res Rev1992 46 6131ndash61401996 58 23ndash30103 h grille k karch and f bechstedt Physica B 1996

133 g s nolas v g harris t m tritt and g a slack J Appl220 690ndash692Phys 1996 80 6304ndash6308104 r s prasher and p e phelan J Heat T ransf 1998 120

134 j-p fleurial a borshchevsky t caillat d t morelli1078ndash1086and g p meisner Proc 15th Int Conf on lsquoThermoelectricsrsquo105 b yang and g chen Phys Low-Dimensional Struct J 20001996 91ndash95 (IEEE Cat no 96TH8169)5 6 37ndash48

135 b c sales d mandrus and r k williams Science 1996106 j d walker d m kuchta and j s smith Appl Phys Lett22 1325ndash13281991 59 2079ndash2081

136 d j braun and w jeitschko J Less-Common Met 1980107 l i arutyunyan v n bogomolov n f kartenko d a72 147ndash156kurdyukov v v popov a v prokofrsquoev i a smirnov and

137 m fornari and d j singh Appl Phys Lett 1999 74n v sharenkova Phys Solid State 1997 39 510ndash5143666ndash3668108 d w song w-n shen t zeng w l liu g chen b dunn

138 h takizawa k miura m ito t suzuki and t endoc d moore m s goorsky r radetic and r gronskyJ Alloy Compd 1999 282 79ndash83ASME HTDndashVol 364ndash1 1999 339ndash344

139 h uchida v crnko h tanaka a kasama and k matsub-109 s g volz and g chen Appl Phys Lett 1999 75 2056ndash2058ara Proc 17th Int Conf on lsquoThermoelectricsrsquo ICTrsquo98 1998110 s g walkauskas d a broido k kempa and t l reinecke330 (IEEE Cat no 98TH8365)J Appl Phys 1999 85 2579ndash2582

140 t caillat j-p fleurial g j snyder a zoltan d zoltan111 p kim l shi a majumdar and p l mceuen Phys Revand a borshchevsky Proc 34th Intersociety EnergyLett 2001 87 215502-1ndash5502-4Conversion Engineering Conf IECE99CD 1999ndash01ndash2567112 t caillat a borshchevsky and j-p fleurial Proc 7th1ndash7Int Conf on lsquoThermoelectricsrsquo (ed K Rao) Arlington TX

141 c cros m pouchard p hagenmuller and j s kasperUniversity of Texas 98ndash101 1993Bull Soc Chim Fr 1968 7 2737113 j-p fleurial t caillat and a borshchevsky Proc 13th

142 j gallmeier h schaoslashffer and a weiss Z Naturforsch 1969Int Conf on lsquoThermoelectricsrsquo 209ndash211 1994 New York24B 665AIP Press no 316

143 g s nolas g a slack and s b schujman Semicond114 c uher Semicond Semimet 2001 69 139ndash253Semimet 2001 69 255ndash300115 s rundqvist and n-o ersson Arkiv for K emi 1968 30

144 g s nolas j l cohn g a slack and s b schujman Appl103ndash114Phys Lett 1998 73 178ndash180116 w jeitschko and d braun Acta Crystallogr 1977 B33

145 b c sales b c chakamoukos d mandrus and j w sharp3401ndash3406J Solid State Chem 1999 146 528ndash532117 j-p fleurial t caillat and a borshchevsky Proc 16th

146 t caillat j-p fleurial and a borshchevsky J PhysInt Conf on lsquoThermoelectricsrsquo 1997 1ndash11 (IEEE Cat noChem Solids 1997 58 1119ndash112597TH8291)

147 s-g kim i i mazin and d j singh Phys Rev B 1998 57118 g p meisner m s torikachvili k n yang m b maple6199ndash6203and r p guertin J Appl Phys 1985 57 3073ndash3075

148 t caillat j-p fleurial and a borshchevsky MRS Symp119 d jung m h whangbo and s alvarez Inorganic ChemProc 478 103 1997 Warrendale PA Materials Research1990 29 2252ndash2255Society

120 f grandjean a gersquorard d j braun and w jeitschko149 t caillat j-p fleurial g j snyder a zoltan d zoltanJ Phys Chem Solids 1984 45 877ndash886

and a borshchevsky Proc 17th Int Conf on121 n t stetson s m kauzlarich and h hope J Solid State

lsquoThermoelectricsrsquo ICTrsquo99 473ndash476 (IEEE Cat noChem 1991 91 140ndash147

99TH8407)122 d j braun and w jeitschko J Less-Common Met 1980

150 s j poon Semicond Semimet 2001 70 37ndash7676 33ndash40 151 c uher j yang s hu d t morelli and g p meisner

123 m e danebrock c b h evers and w jeitschko J Phys Phys Rev B 1999 59 8615ndash8621Chem Solids 1996 57 381ndash387 152 b a cook j l harringa z s tan and w a jesser Proc

124 d t morelli t caillat j-p fleurial a borshchevsky 15th Int Conf on lsquoThermoelectricsrsquo ICTrsquo96 1996 122ndash127j vandersande b chen and c uher Phys Rev B 1995 (IEEE Cat No 97TH8169)51 9622ndash9628 153 m g kanatzidis Semicond Semimet 2001 69 51ndash100

125 t caillat a borshchevsky and j-p fleurial J Appl 154 r t littleton t m tritt c r feger j kolis m lPhys 1996 80 4442ndash4449 wilson and m marone ApplPhys Lett 1998 72 2056ndash2058

126 a watcharapasorn r c demattei r s feigelson 155 t borca-tasciuc r kumar and g chen Rev Sci Instrt caillat a borshchevsky g j snyder and j-p fleurial 2001 72 2139ndash2147J Appl Phys 1999 86 6213ndash6217 156 i hatta Int J T hermophys 1990 11 293ndash303

127 t caillat j-p fleurial and a borschevsky J Cryst 157 t borca-tasciuc w l liu j l liu k l wang and g chenGrowth 1996 166 722ndash726 Proc 35th Nat Heat Transfer Conf NHTCrsquo01 paper no

128 h anno k matsubara y notohara t sakakibara and NHTC2001ndash20097 2001 New York ASME Pressh tashiro J Appl Phys 1999 86 3780ndash3786 158 g chen b yang w l liu t borca-tasciuc d song

129 r p chasmar and r j stratton Electron Contrib 1959 d achimov m s dresselhaus j l liu and k l wang7 52 Proc 20th Int Conf on lsquoThermoelectricsrsquo ICTrsquo01 2001

130 d j singh i i mazin s g kim and l nordstrom MRS 30ndash34 (IEEE Cat no 01TH8589)Symp Proc 478 187 1997 Warrendale PA Materials 159 b yang j l liu k l wang and g chen Appl Phys Lett

2002 80 1758ndash1760Research Society



the units of inverse Kelvin and it often appears as aproduct with an absolute temperature T such as theaverage device temperature Thus the dimensionlessnumerical gure of merit Z T is often cited ratherthan Z by itself

The central issue in thermoelectrics materialsresearch is to increase Z T The best Z T materials arefound in heavily doped semiconductors Insulatorshave poor electrical conductivity Metals have rela-tively low Seebeck coeYcients In addition the ther-mal conductivity of a metal which is dominated byelectrons is in most cases proportional to the electri-cal conductivity as dictated by the WiedmannndashFranzlaw It is thus hard to realise high Z T in metals Insemiconductors the thermal conductivity has contri-butions from both electrons k

eand phonons k

p with

the majority usually coming from phonons Thephonon thermal conductivity can be reduced withoutcausing too much reduction in the electrical conduct-ivity A proven approach to reduce the phononthermal conductivity is through alloying2 The massdiVerence scattering in an alloy reduces the latticethermal conductivity signi cantly without muchdegradation to the electrical conductivity The com-mercial state of the art thermoelectric cooling mater-ials are based on alloys of Bi

2Te

3with Sb

2Te

3(such

as Bi0 middot5

Sb1 middot5

Te3 p-type) and Bi

2Te

3with Bi

2Se

3(such

as Bi2Te

2 middot7Se

0 middot3 n-type) each having a Z T at room

a cooler b power generator c actual device temperature approximately equal to 1 Refrigerators1 Illustration of thermoelectric devices based on such materials typically have a coeYcient

of performance (COP) of about 11 compared tocompressor based refrigerators with a COP between

conductor elements are interconnected on the cold 2 and 4 operating over a comparable working temper-and the hot sides such that a current ows through ature range Their low COP has limited thermo-all the elements in series while the energy they carry electric coolers to niche market sectors such as(by electrons and holes) leaves the cold side in parallel temperature stabilisation of semiconductor lasers andThermoelectric power generators work in reverse to picnic coolers The market for thermoelectric coolersthermoelectric coolers as shown in Fig 1b Because however is rapidly increasing partly due to thethe hot side has a higher temperature electrons and explosive growth of optical telecommunication Stateholes are driven to the cold side through diVusion of the art power generation materials are PbTe andand ow through an external load to do useful work Si

0 middot8Ge

0 middot2 which have been used in deep space radio-

Practical devices are made of many pairs of pndashn legs isotope thermoelectric power generators that operate(Fig 1c) usually arranged such that current ows in at ~900degC with a maximum eYciency of about 7series through all the legs and energy ows in parallel From equation (3) one can infer that the bestfrom the cold side to the hot side thermoelectric devices should have a thermal conduct-

In addition to the temperatures of the hot and cold ivity close to zero One possibility is using vacuumsides which are important to all thermal engines the between the cold and the hot side Electrons can beeYciency of actual thermoelectric devices is deter- emitted through a thermionic emission process frommined by the thermoelectric gure of merit a metal surface and ow through a vacuum This is

the principle behind thermionic power generatorsZ =

S2s

k (3) which were developed in the 1950s3 In a vacuum

based thermionic power generator the emitter is heldat a high temperature Electrons with energy higherwhere s is the electrical conductivity and k is the

thermal conductivity The appearance of S in Z is than the work function can escape from the emittersurface and reach the collector Conceivably ratherself-explanatory The reason that the electrical con-

ductivity s enters Z is due to Joule heating When a than for power generation vacuum thermionic emis-sion can also be used for cooling if a current drivescurrent passes through the thermoelectric elements

Joule heat is generated which can be conducted back electrons from the emitter to the collector as in avacuum tube The major problem however is thatto the cold junction The thermal conductivity k

appears in the denominator of Z because in thermo- most metals have a large work function value whichmakes room temperature refrigeration based onelectric coolers or power generators the thermoelec-

tric elements also act as the thermal insulation vacuum thermionic emission impractical4The progress since the 1960s in improving Z T hadbetween the hot and the cold sides A high thermal

conductivity causes too much heat leakage between been very slow before the 1990s The value of maxi-mum Z T had essentially remained around 1 andthe hot and the cold sides The gure of merit Z has









e=

L( 2 )

e2 Tshy

L( 1 )L( 1 )

e2 TL( 0 ) S=shy

L( 1 )

eTL( 0 )(4)


L(a) =e2 Pd 3k








3 DT =

[(5F3 2 3F

1 2) shy j]2 (3F

1 2 2 )

1B3 D+7F

5 2 2 shy (25F 23 2 6F

1 2)


3 D=

(m )3 2

3p2 A2kB

T

ograve 2 B3 2 k2BTm

ekp


and m= (mxm

ym






BT and F



Fi(j )= P

0

xi dx












1 ndash xGe
















kp= aelig

pP d 3k

8p3[v

p x(k)]2

ccedil fp

ccedil Tt

pograve v


=1

3 P C(v)vp(v)L



pis the phonon



pis










2Te



2Te














2 D Theoretical





























Sbx


























2Te

3to




















1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te































































































e=

L( 2 )

e2 Tshy

L( 1 )L( 1 )

e2 TL( 0 ) S=shy

L( 1 )

eTL( 0 )(4)


L(a) =e2 Pd 3k








3 DT =

[(5F3 2 3F

1 2) shy j]2 (3F

1 2 2 )

1B3 D+7F

5 2 2 shy (25F 23 2 6F

1 2)


3 D=

(m )3 2

3p2 A2kB

T

ograve 2 B3 2 k2BTm

ekp


and m= (mxm

ym






BT and F



Fi(j )= P

0

xi dx












1 ndash xGe
















kp= aelig

pP d 3k

8p3[v

p x(k)]2

ccedil fp

ccedil Tt

pograve v


=1

3 P C(v)vp(v)L



pis the phonon



pis










2Te



2Te














2 D Theoretical





























Sbx


























2Te

3to




















1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te




























































































1 ndash xGe
















kp= aelig

pP d 3k

8p3[v

p x(k)]2

ccedil fp

ccedil Tt

pograve v


=1

3 P C(v)vp(v)L



pis the phonon



pis










2Te



2Te














2 D Theoretical





























Sbx


























2Te

3to




















1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te

































































































2 D Theoretical





























Sbx


























2Te

3to




















1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te































































































Sbx


























2Te

3to




















1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te







































































































1 ndash xSb





1 ndash xSb

xalloys












Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te


































































































Sbx


Bi1 ndash x

Sbx


1 ndash xSb

xis a


1 ndash xSb




Sbx


































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te























































































































































3Sb

2Te





















3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te






























































































3Sb

2Te











kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te



























































































kp=

1

4p Pvmax

0GP 2p

0

sin2 Q dQCPp

0














3Sb

2Te

3and CoSb

3IrSb

3superlattices can






























2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te



































































































2Te



4Sb
















3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te



























































































3) contains square



4]4 ndash


4]4 ndash =4 3


4Pn

1 2 where h is the



4Pn






eleven h T4Pn













1 2(CoSb




2Sb

1 2(Fe

0 middot5Pb

0 middot5Sb

3) If instead Sb is


6Te


1 middot5Te



4Sb

8Te

4(FeSb

2Te) is obtained







4P

1 2 the rare earth



4P

1 2 UFe

4P

1 2and CeFe

4As





3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te


























































































3result in a 70




2Te

3alloys


2Te Fe

0 middot5Ni

0 middot5Sb

3and

Ru0 middot5

Pd0 middot5

Sb3


3and RhSb

















fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te




























































































fFe

4 ndash xCo

xSb


4 ndash xCo

xSb


41 3 4 1 3 5



4 ndash xCo

xSb











Co4Sb


sitions CefFe

4 ndash xNi

xSb






fFe

4 ndash xNi

xSb












8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te


























































































8Ga

1 6Ge











4Sb



4Sb



2Te



8Ga

1 6Ge



4Sb


into c-Zn4Sb



1 6Ge

3 0polycrys-





2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te


























































































2Te



4Sb

3forms a full range


4Sb



CdxSb

3mixed crystals


4Sb



CdxSb





4Sb

3into




2Te





0were


6 middot3 3S

1 0and K

2Bi

8S


K2Bi

8Se

1 3and K

2 middot5Bi

8 middot5Se

1 4 A

1 + xPb

4 ndash 2 xBi

7 + xSe











4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te
































































































4Te



4Te



3 BiI

3 and In

2Te








2Te









Si0 middot8

Ge0 middot2



5and ZrTe






































2Te

3Sb

2Te













































































































2Te

3Sb

2Te
























































































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