quantum transport thermoelectric properties inas/gasb … · 2019. 8. 1. · figure shows our...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 501-505Reprints available directly from the publisherPhotocopying permitted by license only

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Printed in India.

Quantum Transport and Thermoelectric Propertiesof InAs/GaSb Superlattices

J.-F. LIN and D. Z.-Y. TING*

Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 300, ROC

In recent years, artificially layered microstructure have been considered as candidatesfor better thermoelectrics. In this work we examine transport properties of the type-IIbroken-gap InAs/GaSb superlattice. We use the effective bond orbital model for anaccurate description of the band structures. Theoretical results of thermoelectrictransport coefficients and the dimensionless figure of merit for an (8, 8)-InAs/GaSbtype-II superlattice are presented.

Keywords." Thermoelectric, type-II superlattice, superlattice transport, InAs, GaSb

1. INTRODUCTION

From about 1940 to 1965, many scientists madetremendous effort looking for superior thermo-electric materials. The advantage of using thesesolid state devices are compactness, quietness (nomoving parts), freedom from corrosion, localizedheating or cooling, and reliability. The effortdwindled eventually because performance of mostthermoelectric materials found at that time weretoo poor to be used for practical commercialapplications. Recently, there is a renewed interestin this area [1, 2], driven by the following reasons:First, thermoelectric technology is environmen-tally cleaner than traditional compressor-basedrefrigeration technology since it does not usechlorofluorocarbons. Second, several new materi-

als were identified as potential candidates forbetter thermoelectrics, including the filled skutter-udite antimonides [3] and PbTe/Pbl_xEuxTe mul-tiple-quantum-well structures [4]. Among the newmaterials, superlattices attracted many scientists’attention [5, 6, 7, 8, 9]. The interest can be tracedback to the quantitative results first obtained byHicks and Dresselhaus [5] where huge enhance-ment of thermoelectric properties was predictedfor superlattice structures.

In the present work, we focus our interest on thetype-II broken-gap InAs/GaSb superlattices.Superlattices consisting of combinations of III-V semiconductors with type-II band alignmentsare of interest for infrared applications, includingIR detectors [10] and laser diodes [11, 12]. This isbecause their energy gaps can be made smaller

* Corresponding author.

501

502 J.-F. LIN AND D. Z.-Y. TING

than their constituents. However, for most of theseapplications, cooled operation are desirable forgood performance. If reasonable thermoelectricproperties of such superlattice structure could beobtained, directly integration of thermoelectriccooling devices with a set of IR detectors or laserdiodes may offer some advantage. Our aim here isto present the early results of our ongoingtheoretical effort to identify and characterize thethermoelectric properties of the type-II broken-gap InAs/GaSb superlattices. In Sec. II, thetheoretical framework for calculating electroniccontributions to the thermoelectric properties ofsuperlattices using realistic band structure modelsis presented. Our results for an (8, 8)-InAs/GaSbsuperlattice is presented in Sec. III.

2. MODEL

The dimensionless figure of merit for a thermo-electric material is given by [14, 15]

S 2crTzr=

where S is the thermopower (thermoelectric poweror Seebeck coefficient), a is the electrical con-ductivity, is the total thermal conductivity(=+ he, the lattice and electronic contribu-tions, respectively), and T is the temperature. Themaximum Coefficient Of Performance (COP) isdirectly related to the dimensionless figure of meritof the material by [14, 15]

COP-- Tc v/1 4- ZT- Th/Tc, (2)Th- Tc v/1 4-ZT 4-

where Tc, Th are the temperatures at the cold andhot junctions, respectively, and T, which is equalto (Th + Tc)/2, is the mean absolute temperature,for a single-stage thermoelectric cooler. As ZTgoes to infinity, the COP goes to the thermo-dynamic limit of Carnot efficiency.

The low-field transport coefficients for thermo-electric materials are defined by

J erE aSVT (3)

JQ TerSE- toTT (4)

where J is the electric current density, JQ thethermal current density, E the electric field, Tthe temperature, cr the electrical conductivity, Sthe thermopower, and n0 the thermal conductivityat zero electric field. In general, a, S, and t0 are3 3 matrices in the Cartesian coordinates. Notethat the value of needed for computing Z is thetotal heat flow for zero electrical current, so it isgiven by

n, n,o To’S 2 @ i,, (5)

where nt is the lattice contribution to the thermalconductivity.

Using the relaxation time approximation to theBoltzmann equation, the transport coefficients aregiven by [16]

f (Of)(e) (6)o--e 2 de ----eJL ( Of)!,(e)(e--#) (7)TaS -e d j

To de -Nj(e)(e- #)2 (8)

where is the chemical potential, e the electroncharge, and f0 the Fermi-Dirac distribution func-tion. Here (e) is given by:

(e) -(e) Z v(k)v(k)(e- (k)) (9)k

where v(k) Vke(k)/h is the semi-classical carriervelocity, e(k) is the dispersion relation for thecarriers, and -(e) is the energy-dependent relaxa-tion time tensor, taken to be k-independent. Thesummation is over the first Brillouin zone.Equations (6), (7), and (8) are the basic results ofthe theory of electronic contributions to the

THERMOELECTRIC PROPERTIES InAs/GaSb SUPERLATTICES 503

thermoelectric effects. It should be noted thatphonon contributions are ignored here.The theoretical scheme we used for calculating

of the electronic structure of InAs/GaSb super-lattices is the Effective Bond-Orbital Model(EBOM) [13]. This method combines the virtuesof the k.p and the tight-binding methods. Thebasic idea is to use a minimum number of bondorbitals to describe, as accurately as possible, themost relevant portion of the bulk band structures,and then use them in a supercell calculation toobtain superlattice band structures. In our case,eight bond orbitals per unit cell, including the s-like bond orbitals with spin up and spin down,four bond orbitals each with total angularmomentum J--- 3/2 (made ofp-like states coupledwith spin) and two additional bond orbitals eachwith total angular momentum J-- 1/2 (also madeofp-like states coupled with spin), are used. This isbecause the superlattice states of interest containadmixture of both valence-band and conduction-band characteristics. We assume that all the bondorbitals are sufficiently localized so that theinteraction between orbitals separated farther thanthe nearest neighbor distance can be ignored. Allnonvanishing interaction parameters can then bedirectly related to the effective masses or otherband parameters of the k.p perturbation theory.Based on this model, an accurate band structurecould be obtained for values of k near the zonecenter.Our strategy for computing thermoelectric

properties is as follows. First, we calculate thesuperlattice band structures by using SOBOmodel. Then, we perform full Brillouin zoneintegration to obtain (). Finally, we use Equa-tions (6), (7) and (8) to compute the transportcoefficients.

3. RESULTS AND DISCUSSION

Figure shows our calculated band structure foran InAs/GaSb superlattices eight monolayers ofInAs and eight monolayeres of GaSb per period.

1.5

1.0

0.5

Band Structure(8,8) InAs/GaSb Superlattice

-0.50.05 0.00 0.05 0.10 0.15 0.:20

[001 Wave Vector (2n/a) [100]

FIGURE Illustration of band structure of (4, 4) InAs/GaSbsuperlattices. Note that in our calculation the strain effect isincluded.

The conduction band minimum of bulk InAs istaken to be at 0 eV. Experimental band offsetvalue [17] between InAs and GaSb then puts thevalence band top of GaSb at approximately 200meV above the conduction band minimum of theInAs conduction band edge. In the figure, super-lattice band structure perpendicular and parallel togrowth directions are shown. The valence subbandmaximum is found at 0.086 eV, while the conduc-tion subband minimum at 0.303 eV, yielding asuperlattice bandgap of 0.217 eV. Note that theconduction subband structure along the super-lattice growth direction is still very dispersive. Thisis due the broken-gap band alignment, the smallInAs conduction band effective mass, and therelatively short period.To compute thermal electric properties, we need

to know relaxation times. Lacking specific knowl-edge of relaxation times in superlattice structures,we made an estimate from the mobility datafor bulk InAs and GaSb, and assume a value of

504 J.-F. LIN AND D. Z.-Y. TING

"r(e) 10-13 sec. All of our calculations assume atemperature of 300 K. In Figure 2, we plot thethermoelectric transport coefficients as functionsof the chemical potential. Due to the fact thattransport along the growth (transverse) direction isimpeded by scattering from the superlattice inter-faces, the transverse components (zz) of electricalconductivity (or) and electronic contribution tothermal conductivity (he) are always smaller thanthe parallel (xx) components. The transversecomponents of all three transport coefficients showstrong oscillations as functions of chemical poten-tial. This can be understood by examining thesuperlattice minibands along the transverse ([001])direction. We can easily see that the oscillationstrack the miniband edges.

In order to calculate the thermoelectric figure ofmerit for this structure we need the latticecontribution to the thermal conductivity. Fromthe experimental data of bulk InAs and GaSb, wetake the lattice thermal conductivity of superlatticeas the average of two bulk values, which isapproximately 35 W/K m. This procedure prob-ably overestimates the values of lattice conductiv-

ity since we would expect interface scattering tolower thermal conductivities. In Figure 3, we showthe dimensionless figure of merit ZTvs. thechemical potential. For parallel transport, theoptimum value occurs when chemical potential isabout 0 eV. Its value is about 0.033 which is notbetter than the bulk value. This result is oppositeto that of Hicks’ calculation [5] where ZT for atype-I superlattices better than bulk materials. Thedifference is probably due to the much strongerinter-well coupling in the type-II broken gapsuperlattices.While our calculations have carefully treated

band structure effects, the models we used toestimate relaxation times and lattice thermalconductivities have been rather crude. In particu-lar, it is likely that we considerably over-estimatedthe superlattice thermal conductivity [2]. Theseissues must be addressed before we could obtain amore realistic description of superlattice thermo-electric properties. In summary, we have calcu-lated the thermoelectric properties, includingconductivity, electronic contribution to thermalconductivity, and thermal power, for a type-II

Transport Coefficients(8,8)-InAs/GaSb Superlattice

107

10

10

"; 104103

102102

10

100

1(R)10

10

4O0

20O

-200

-400

-0.5

Szz

010 0.5 1.0 1.5Chemical Potential (eV)

FIGURE 2 Shown are the conductivity, electronic contribu-tion to the thermal conductivity, and thermopower vs. thechemical potential at T 300 K.

10

0

10

10

0-a_0.5 0’.0

Figure of Merit(8,8)-InAs/GaSb Superlattice

paralleltransverse

i.

:il

l00.5 . 1.5Chemical Potantial (V)

FIGURE 3 ZT vs. chemical potential for the (8,8)-InAs/GaSb superlattices.

THERMOELECTRIC PROPERTIES InAs/GaSb SUPERLATTICES 505

broken-gap InAs/GaSb superlattice using realisticband structure models. The computational toolsdeveloped form a basis for further explorations ofsuperlattice thermoelectric properties.

Acknowledgements

The authors would like to thank C.C. Chi forhelpful discussions. This work was supported bythe ROC National Science Council under GrantNo. NSC 86-2112-M-007-001. The use of ROCNational Center for High-Performance Comput-ing facilities is acknowledged.

References

[1] Tritt, T. M. (1996). "Thermoelectrics run hot and cold",Science, 272, 1276-1277.

[2] Mahan, G., Sales, B. and Sharp, J. "Thermoelectricmaterials: new approaches to an old problem", PhysicsToday, March 1997, 42-47.

[3] Sales, B. C., Mandrus, D. and Williams, R. K. (1996)."Filled skutterudite antimonides: a new class of thermo-electric materials", Science, 272, 1325-1328.

[4] Hicks, L. D., Harman, T. C., Sun, X. and Dresselhaus, M.So (1996). "Experimental study of the effect of quantum-well structures on the thermoelectric figure of merit",Phys. Rev., B53, 10493-10496.

[5] Hicks, L. D. and Dresselhaus, M. S. (1993). "Effect ofquantum-well structures on the thermoelectric figure ofmerit", Phys. Rev., B47, 12727-12731.

[6] Sofo, J. O. and Mahan, G. D. (1994). "Thermoelectricfigure of merit of superlattices", Appl. Phys. Lett., 65 (21),2690-2692.

[7] Mahan, G. D. and Lyon, H. B. (1994). "Thermoelectricdevices using semiconductor quantum wells", J. Appl.Phys., 76(3), 1899-1901.

[8] Whitlow, L. W. and Hirano, T. (1995). "Superlatticeapplications to thermoelectricity", J. Appl. Phys., 78 (9),5460- 5466.

[9] Lin-Chung, P. J. and Reinecke, T. L. (1995). "Thermo-electric figure of merit of composite superlattices systems",Phys. Rev., B51, 13244-13248.

[10] Chow, D. H., Miles, R. H., Schulman, J. N., Collins, D.A. and Mcgill, T. C. "Type-II superlattices for infrareddetectors and devices", Semi. Sci. Tech., v6(12C), 47-51(1991 Dec.).

[11] Chow, D. H., Miles, R. H., Hasenberg, T. C., Kost, A. R.,Zhang, Y. H., Dunlap, H. L. and West, L. "Mid-waveinfrared diode-lasers based on GaInSb/InAs and InAs/A1Sb superlattices", Appl. Phys. Let., 67 (25), 3700-3702(1995 Dec. 18).

[12] Miles, R. H., Chow, D. H., Zhang, Y. H., Brewer, P. D.and Wilson, R. G. "Midwave infrared stimulated-emissionfrom a GaInSb/InAs superlattices", Appl. Phys. Let.,66(15), 1921-1923 (1995 April. 10).

[13] Chang, Y.-C. (1988). "Bond-orbital models for super-lattices", Phys. Rev., B37, 8215- 8222.

[14] Ioffe, A. F. Semiconductor Thermoelements and Thermo-electric Cooling (Infosearch Ltd., 1957).

[15] Goldsmid, H. J. Applications of Thermoelectricity (Lon-don: Methuen & Co Ltd. New York: John Wiley & SonsInc., 1960).

[16] Ashcroft, N. W. and Mermin, N. D. Solid State Physics,Chapter 13 (Harcourt Brace College Publishers, Interna-tional Edition).

[17] Sai-Halasz, G. A., Chang, L. L., Walter, J. M., Chang, C.A. and Esaki, L. (1978). Solid State. Commun., 27, 935.

Authors’ Biographies

Jie-Feng Lin received a B.S. degree in Electro-physics from the National Chiao Tung University,Taiwan in 1995, and an M.S. Degree in Physicsfrom the National Tsing Hua University, Taiwanin 1997. His involves theoretical studies ofsemiconductor heterostructures.

David Z.-Y. Ting is an Associate Professor ofPhysics at the National Tsing Hua University inHsinchu, Taiwan, ROC, and a Visiting ResearchAssociate at the California Institute of Technol-ogy. His research activities include theoreticalstudies of electronic and optical properties ofsemiconductor alloys and heterostructures, quan-tum transport in nanostructures and tunneldevices, and optical simulations.

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