PHYSICAL REVIEW B 1 NOVEMBER 1997-IVOLUME 56, NUMBER 17

Phonon superlattice transport

Per Hyldgaard* and G. D. MahanDepartment of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200

and Solid State Division, P.O. Box 2008, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6030~Received 9 July 1997!

We predict in Si/Ge superlattices a dramatic high-temperature suppression of the perpendicular thermaltransport. Total internal reflection confines the superlattice modes and significantly reduces the average groupvelocity at nonzero in-plane momenta. These consequences of the acoustic mismatch cause at high tempera-tures an order-of-magnitude reduction in the ratio of superlattice thermal conductivity to phonon relaxationtime. @S0163-1829~97!07441-9#

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The observed1,2 reduction of the in-plane thermal condutivity in long-period GaAs/AlAs superlattices implies a ptentially attractive application for semiconductor heterostrtures in thermoelectric devices.3 The in-plane superlatticethermal transport has consequently been the focus of theical investigations.4,5

Two recent experiments6,7 also report a dramatic reduction for the thermal transport perpendicular to tGaAs/AlAs superlattice interfaces. The perpendicular thmal conductivitykSL reported in Ref. 7 exhibits an order-omagnitude room-temperature reduction compared withof bulk GaAs and cannot be explained as alloy scatterin

A very recent paper8 investigates this microscopic hetransfer and reports a Boltzmann transport calculation baon abulk phonon dispersion in each of the superlattice laers. Assuming complete diffusive scattering9,4,5 at every in-terface, that is, a very strong reduction in the effective pnon relaxation timetSL , it is possible to account for theobserved6,7 reduction in the perpendicular superlattice thmal conductivity.8

We believe it essential, however, to describe the supetice thermal transportkSL across the increasingly perfect interfaces based instead on thesuperlattice-phononspectrum.Total internal reflection results with a finite acoustic mmatch difference in material sound velocities in a significsuppression of the phonon flux across an individual hetestructure interface.10 We must therefore expect a corresponing reduction in the perpendicular group velocity of the sperlattice phonon modes, that is, an effective moconfinement. Below we document how the acoustic mmatch causes such a modal confinement and producesperlattice phonon spectrum qualitatively different from thoof the individual bulk layers.

Early one-dimensional model investigations yielded phnon spectra for GaAs/AlAs~Refs. 11 and 12! and Si/Ge~Ref.13! superlattice modes propagating at right angles to theterfaces, i.e., with zero in-plane momentum,q[0. Suchmodes can be probed optically and as a function of perpdicular momentumk using ultrasound.11 The increased umklapp scattering of the zone-foldedq50 GaAs/AlAs super-lattice modes causes a small (,25%) reduction in theeffective phonon relaxation timetSL and thus in the thermaconductivity.14

More recently there have been several three-dimensicalculations of the superlattice phonons identifying both

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effects of strain15 and a finite interface disorder.16 Most in-vestigations report the dispersion,v(q,k), of superlatticemodes propagating either entirely within the superlattplanes,k[0, or at right angles to the interfaces,q[0. Noneof these studies, however, directly illustrate the effectmodal confinement,]v(k;q)/]k'0, arising atqÞ0.

In this paper we attempt such an illustration and alsoplore the important consequences for the perpendicular tmal conductivitykSL of Si/Ge superlattices. We assume pefect interfaces and use a simple-cubic effective phonmodel17 to ~i! document how the finite acoustic mismatcensures a dramaticqÞ0-modal confinement, and~ii ! predicta resultingorder-of-magnitudehigh-temperature suppressioof the perpendicular thermal transport. This reduction aridirectly in the ratiokSL /tSL and thus our work complementearlier investigations8,14 of the effective superlattice phonorelaxation timetSL . We emphasize that the~smaller! acous-tic mismatch will drive a similar~but smaller! modal con-finement and thermal conductivity reduction also in tGaAs/AlAs superlattices.6,7

Figure 1 characterizes our simple-cubic model of tdouble-silicon/double-germanium superlattice phononnamics. The upper panel shows one-dimensional moschematic assuming effective silicon/germanium lattice cstants aSi/Ge and atomsMSi/Ge connected by intrasiliconintragermanium force constantsFp;Si/Ge and interlayer cou-pling constantsKp[AFp;SiFp;Ge.

18 This one-dimensionamodel, however, can only describe the dynamics of supetice phonons both polarized (jpi z) and propagating exactlyin the growth orz direction.

The lower panel of Fig. 1 illustrates our simple-cubmodel of the general superlattice phonon dynamics. Wesume for simplicity a shared in-plane and perpendiculartice constant,a[aSi ~identical in all layers!, and add in-plane force constants19 Ft;Si/Ge necessary to describe thphonon propagation at a finite in-plane momentumqÞ0. Wealso introduce characteristic frequenciesVp,t;Si~Ge!

[A4Fp,t;Si~Ge! /MSi~Ge!.Thejpi z-polarized phonon modes at in-plane momentaq

is then described by a bulk silicon~germanium! model dis-persion:

vq25Vp;Si~Ge!

2 @12cos~kSi~Ge!a!#/21V t;Si~Ge!2 ~aq/2!, ~1!

aq5@22cos~qxa!2cos~qya!#,4, ~2!

10 754 © 1997 The American Physical Society

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56 10 755BRIEF REPORTS

relating frequencyvq and perpendicular momentumkSi~Ge! .We describe the bulk silicon~germanium! modes with polar-ization j x,yi x,y using the dispersion

vq,x,y2

5V t;Si~Ge!2 @12cos~kSi~Ge!a!#/2

1~Vp;Si~Ge!2 1V t;Si~Ge!

2 !~aq/4!. ~3!

Silicon is significantly harder than germanium. Our simpmodel yields Vp;Si~Ge!540.9 ~22.9! meV and V t;Si~Ge!528.3 ~16.5! meV and is not an accurate description of tlongitudinal or transverse phonon modes.20 However, the ra-tios Vp,t;Si

2 ;3Vp,t;Ge2 are consistent with the actual bulk ph

non spectra and our model thus reflects the essential efof the acoustic mismatch. Below we concentrate our discsion on thejp-polarized modes.

Assuming perfect interfaces the in-plane momentaq isconserved and thus characterizes also the superlattice mtogether with the squared frequency valuevq

2. The perpen-dicular dynamics of such a mode is described byindividual-layer model dispersion

vq;Si/Ge2 5Vp;Si/Ge

2 @12cos~kSi/Gea!#/2, ~4!

FIG. 1. Simple-cubic model for double-silicon/doublgermanium superlattice phonon spectra. The upper panel shone-dimensional model schematic with silicon/germanium atoMSi/Ge and lattice constantsaSi/Ge. The atoms are coupled bintrasilicon/intragermanium force constantsFp;Si/Ge and interlayerconstantKp[AFp;SiFp;Ge. The lower panel illustrates the simplecubic phonon model for the generaljpi z-polarized superlatticemodes, that is, having anonzeroin-plane momentum,qÞ0. Weassume a shared in-plane and perpendicular lattice constaa[aSi ~identical in all layers,! and add in-plane silicon/germaniumforce constants,Ft;Si/Ge.

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where vq;Si/Ge2 [vq

22V t;Si/Ge2 (aq/2) defines the effective

squared frequency available for the perpendicular motwithin the silicon/germanium.

A superlattice mode (q,vq2) will propagate~i.e., be char-

acterized by a real wave vectorkSi/Ge! through a silicon/germanium layer if and only if 0<vq;Si/Ge

2 <Vp;Si/Ge2 . In con-

trast, superlattice modes withvq;Si~Ge!2 ,0(.Vp;Ge

2 ) aregermanium~silicon! confined by total internal reflection. Foexample, a high-energy mode withvq;Si

2 ;Vp;Si2 yields a sub-

monolayer decay length 1/gGe as may be estimated from

cosh~gGea![~2vq;Si2 /Vp;Ge

2 !21. ~5!

We describe the general phonon displacement field azcoordinatej in layer l and in-plane momentumq by

hq; l~ j !5Alexp~ ik la j !1Blexp~2 ik la j !, ~6!

using either real or complex wave vectorskl 5Si/Ge for modeswhich in the silicon/germanium are propagating or evancent, respectively. For thejpi z-polarized modes theindividual-layer displacement fields, Eq.~6!, couple at theheterostructure interfaces through force constantKp

[AFp;SiFp;Ge. The equation of motion for the two atombordering the l / l 11 interface establishes the relatio(Al ,Bl)↔(Al 11 ,Bl 11).12

Our four-atom superlattice has periodd54a and is de-scribed by the perpendicular superlattice phonon momenkP@2p/d,p/d#. We solve the equations linking(Al ,Bl)↔(Al 12 ,Bl 12) subject to the Bloch ansatz atk:

hq; l 12~ j 14!5hq; l~ j !eikd, ~7!

to obtain the frequency-squared values,vq;k,l2 , l5124, and

thus determine the~zone-folded! superlattice modal spectr~for jpi z- as well as forj x,yi x,y-polarized modes.!

We assume a frequency-independent phonon relaxatime tSL and denote byN(T) the Bose-Einstein distributionfunction at temperatureT. Generalizing the Boltzmann equation result for bulk phonon transport21 we evaluate the ratioof the perpendicular superlattice thermal conductivitykSL tophonon relaxation timetSL using

kSL

tSL~T!5\(

mF E d2q

~2p!2 E2p/d

p/d dk

2p

3(l

vq;k,lS ]vq;k,l

]k D 2S dN

dTD Gm

. ~8!

Here the indexm denotes a sum over polarization (jp ,j x,y)and l identifies the set of superlattice modes atm, q, and~zone-folded! perpendicular momentumk. The correspond-ing estimates,kSi/Ge/tSi/Ge, for bulk silicon/germanium re-sult by replacing in Eq.~8! d by aSi/Ge and of course restrict-ing l to the single acoustic band described by our effectsimple-cubic lattice model. Note that we divide out the vedifferent phonon relaxation times,tSi/Ge/SL, to obtain a faircomparison of the thermal conductivities and illustratesuperlattice effects.

Figure 2 compares the temperature variation of supertice ratio kSL /tSL, solid curve, with the corresponding estmates for bulk silicon, dashed curve, and for bulk germ

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10 756 56BRIEF REPORTS

nium, dotted curve. To interpret this temperature variationk/t both for the superlattice and in the bulk we approximk as a product of the average group velocity,^]v/]k&, av-erage mean free path,t^]v/]k&, and of the heat capacity,C:

k;t^]v/]k&2C/3. ~9!

In the low-temperature limit we may approximate tbulk silicon/germanium heat capacity,CSi~Ge!}(T/ vSi~Ge!)

3

wherevSi~Ge! denotes the average long-wavelength soundlocity. Thus we have ratioskSi/Ge/tSi/Ge}T3/ vSi/Ge yielding acubic temperature dependence for the bulk materials.low-temperature ratiokSL /tSL is an average ofkSi/Ge/tSi/Gebecause the long-wavelength superlattice velocity is an aage of the Si/Ge velocities as documented in the inset of2.

In contrast, the high-temperature ratiokSL /tSL shows adramatic suppression compared with the average bsilicon/germanium values. This reduction reflects a suppsion of the perpendicular average group velocity:

k/t}^]v/]k&2; T*Vp,t;Si/Ge, ~10!

as the high-temperature bulk silicon/germanium and sulattice heat capacities saturate at the Boltzmann constimes the density of phonon modes. To understand thematic reduction of the high-temperature ratiokSL /tSL we in-vestigate below the zone-folded superlattice phonon spe

FIG. 2. Dramatic suppression of perpendicular superlattice tmal transport. The solid curve shows order-of-magnitude hitemperature reduction for calculated superlattice ratio of therconductivity (kSL) to phonon relaxation time (tSL) when comparedto average of bulk silicon/germanium ratios, dashed/dotted cuThe conductivity suppression results from the nonzero in-planementaqÞ0 ensuring an effective confinement of the superlattmodes~Fig. 3.! The inset verifies that no such confinement effearises in the long-wavelength limit~relevant for the low-temperature transport!: lowest q50 superlattice spectrum, solicurve, is an average of corresponding Si/Ge spectra, dashed/dcurves.

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at general in-plane momentaq and illustrate how the acoustic mismatch forces a significant reduction of the averaphonon group velocity.

Figure 3 compares squared frequency values,vq2, for

zone-folded superlattice modes, (0<kSLd<p) central col-umn, and for bulk-silicon/germanium propagating mode(0<kSi/Gea<p) left/right column, at increasing in-planemomentumq. The bottom row of panels compares thephonon spectra ataq50, the middle row ataq50.9, and thetop row ataq52. All panels shows squared frequencies retive to V t;Si

2 (aq/2) ~dashed lines,! minimal value necessaryfor propagation in the silicon. The pairs of dotted lines idetifies the range of squared frequency values which allogermanium propagation, 0<vq;Ge

2 <Vp;Ge2 .

The superlattice spectrum evaluated ataq50, centralbot-tom panel, resembles the spectra obtained in previous odimensional theoretical studies.12,13 The aq50 dispersioncurves show two modes which can propagate in all lay

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FIG. 3. Increasing silicon/germanium superlattice phonon cfinement for upper/lower two zone-folded modes illustrated ataq

[@22cos(qxa)2cos(qya)#50 ~bottom row!, at aq50.9 ~middlerow!, and ataq52 ~top row!. The central column of the panelshow the squared frequency values,vq

2, of zone-folded superlatticemodes (0<kSLd<p). The left ~right! column of panels compareswith squared frequency values of modes able to propagate in bsilicon ~germanium!, 0<kSi~Ge!a<p. All panels shows squared frequencies relative toV t;Si

2 (aq/2) ~dashed lines!—minimal value nec-essary for silicon propagation ataq . The dotted lines identifiessquared frequencies which ataq allow germanium propagation.

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56 10 757BRIEF REPORTS

and two modes located above the range of germaniupropagating modes. The top superlattice mode is almosttirely siliconlike. It is much too energetic to penetrate tgermanium, and is consequently characterized by a vaning group velocity.13

The intermediate valueaq50.9, centralmiddle panel,forces a finite offset between silicon- and germaniupropagating modes as identified by the set of dasheddotted lines. The offset quantifies the traditional total intereffect: germanium phonons at a givenqÞ0 must have also afinite perpendicular momentum—that is, they must hamore than the minimum energy set byqÞ0—to enter theharder silicon.

In turn this offset between the silicon- and germaniupropagating phonons causes a partial germanium confiment of the lowest superlattice mode and an increased silconfinement of both upper two modes. Already ataq50.9we observe a significantly reduced contribution to the avage superlattice phonon group velocity.

The centraltop panel illustrates how in-plane momenwith aq>2 ensures a partial confinement of all superlattphonon modes. The range of squared frequency valuesmodes propagating in the silicon~germanium! layers is ataq52 situated entirely above~below! the dashed-dotted lineThe top~bottom! two superlattice modes are thus ataq>2

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increasingly silicon-~germanium-! confined even by the thinadjacent germanium~silicon! layers.

The valueaq52 corresponds roughly to a minimal phonon energy of one half the germanium Debye temperatuAt such temperatures and above the average modal confiment causes a dramatic suppression of the ratiokSL /tSL asdocumented in Fig. 2.

We have theoretically investigated the thermal transpperpendicular to the interfaces in a double-silicon/doubgermanium superlattice assuming a simple-cubic modelthe lattice vibrations. We predict an order-of-magnitudhigh-temperature suppression in the ratio of the perpendilar thermal conductivity to the phonon relaxation time. Thdramatic suppression occurs at temperatures above onethe germanium Debye temperature when the acoustic mmatch forces an effective modal confinement at the finin-plane momentaq characterizing this transport. Uponcompletion of this work we have learned of recent Si/Gsuperlattice measurements reporting perpendicular therconductivitiessmaller than that of an actual Si/Ge alloy.22

This work was supported by ARPA, the University oTennessee, and the Division of Material Science, U.S. Dpartment of Energy under Contract No. DE-AC0596OR22464 with Lockheed Martin Energy Research Corpration.

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*Present address: Dept. of Elec. Eng., University of Notre DamNotre Dame, IN 46556.

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18We choose effective lattice constantsaSi/Ge to fit actual mass den-sities, and force constantsFp;Si/Ge to yield simple-cubic soundvelocity vp;Si/Ge5(alAFp; l /Ml) l 5Si/Ge equal to the silicon/germanium values reported for@100# longitudinal phonons in H.H. Landolt and R. Bo¨rstein, Numerical Data and FunctionalRelationships in Science and Technology, New Series, 17e,~Springer-Verlag, Berlin, 1983!, p. 63/397. In the long-wavelength limit this choice of effective parameters is consistenwith the continuum-mechanics-based description of heterostructure phonon transmission probabilities, see for example S. Mizuno and S. Tamura, Phys. Rev. B45, 734 ~1992!.

19The force constantsFt;Si/Ge are fitted to the sound velocities of@100# transversephonons in bulk silicon/germanium.

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