Energy relaxation of two-dimensional carriers in strained Ge/Si 0.4 Ge 0.6 and Si/Si 0.7Ge 0.3 quantum wells: Evidence for two-dimensional acoustic phononsS.-H. Song, Wei Pan, D. C. Tsui, Y. H. Xie, and Don Monroe Citation: Applied Physics Letters 70, 3422 (1997); doi: 10.1063/1.119190 View online: http://dx.doi.org/10.1063/1.119190 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/70/25?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strain effect analysis on phonon thermal conductivity of two-dimensional nanocomposites J. Appl. Phys. 106, 114302 (2009); 10.1063/1.3259383 Observation of two-dimensional hole gas with mobility and carrier density exceeding those of two-dimensionalelectron gas at room temperature in the SiGe heterostructures Appl. Phys. Lett. 91, 082108 (2007); 10.1063/1.2773744 High room-temperature hole mobility in Ge 0.7 Si 0.3 /Ge/Ge 0.7 Si 0.3 modulation-doped heterostructures J. Appl. Phys. 89, 2497 (2001); 10.1063/1.1334632 Temperature dependence of the electron–phonon scattering time of charge carriers in p -Si/SiGe heterojunctions Low Temp. Phys. 26, 890 (2000); 10.1063/1.1334440 Hot hole energy relaxation in Si/Si 0.8 Ge 0.2 two dimensional hole gases J. Appl. Phys. 81, 6853 (1997); 10.1063/1.365244

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Energy relaxation of two-dimensional carriers in strained Ge/Si 0.4Ge0.6and Si/Si 0.7Ge0.3 quantum wells: Evidence for two-dimensionalacoustic phonons

S.-H. Song,a) Wei Pan, and D. C. TsuiDepartment of Electrical Engineering, Princeton University, Princeton, New Jersey 08544

Y. H. Xie and Don MonroeBell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974

~Received 13 March 1997; accepted for publication 20 April 1997!

We performed heating measurements on holes in a strained Ge/Si0.4Ge0.6 quantum well andelectrons in a strained Si/Si0.7Ge0.3quantum well in the temperature range 0.3–5.5 K. While a powerlaw dependence of carrier temperature on current,Te;I a, was observed for both samples, themeasured values for the current exponent are different:a50.5060.02 for the Ge sample and 0.4060.02 for the Si sample. We attribute this exponent difference to the difference in their phonondimensionality. ©1997 American Institute of Physics.@S0003-6951~97!03025-8#

Electron heating experiments have been widely used tostudy the electron–phonon interaction in metal films1 and insemiconductors.2,3 Due to the nature of the experiment, theemission of phonons dominates the interaction. From manytheoretical and experimental studies, the energy relaxationrate (1/te) by phonon emission at temperatures (T) belowthe Debye temperature, is known to follow power-law de-pendence onT:

1/te;Ta. ~1!

For clean bulk metals, it is now believed thata53. Thisresult can be most easily understood if one considers that therate is proportional to the number of phonons with thermalmomenta.4 Thus, theT exponenta simply reflects the di-mensionality of the phonons.

It is expected that the exponent is reduced to 2 for thinmetal films with thickness smaller than the thermal phononwavelength. However, the experimentally obtained valuesfor the exponent vary widely. While the experiment on freestanding films shows the expected exponent of 2,5 the valuesobtained for films on acoustically mismatched substratesrange from 1.3 to 4.0.6 In the latter case, in which most of theexperiments were carried out, the interface between a metalfilm and a substrate can be very different from an ideal in-terface depending on the condition and method of preparingthe film. A rough interface will increase the phonon–phononcoupling to the substrate and thus weaken the effect of thereduced dimensionality of the phonons.

On the other hand, in semiconductor heterostructures,almost ideal, atomically abrupt interfaces can be routinelygrown using advanced crystal growth techniques like themolecular beam epitaxy~MBE!. The exponent of 3 has beenreported from experiments on the two-dimensional~2D!electron system in GaAs–AlGaAs heterostructures at lowT’s.3,7 These two semiconductors are lattice matched to eachother. They do not provide large enough acoustic mismatchto trap the phonons, so the phonons are believed to be 3D.Note that it is the phonon dimensionality that determines theexponent, not the electron dimensionality.

Recent progress in strained layer epitaxy offers a newpossibility of growing lattice mismatched semiconductor ma-terials. Among semiconductor materials, Si and Ge are con-tinuously miscible and can be pseudomorphically grown on alattice mismatched Si substrate. They have very large densitydifference~2.33 g/cm3 for Si and 5.32 g/cm3 for Ge!, andthus may provide a large enough mismatch to confinephonons.

In this letter, we report a heating experiment of 2D car-riers in strained Si/Si0.7Ge0.3 and Ge/Si0.4Ge0.6 quantum wellson ~100! Si substrates. We have measured two samplesgrown by MBE technique, one with holes in a strained Gelayer sandwiched between Si0.4Ge0.6 layers~sample X516, tobe referred to as the ‘‘Ge sample’’! and the other with elec-trons in a strained Si layer between Si0.7Ge0.3 layers~X571,likewise, the ‘‘Si sample’’!. The detailed sample structuresare shown in Fig. 1. One of the primary effects of the strainon the band structure is the removal of degeneracies.8 In theSi sample, the biaxial tensile strain splits the sixfold conduc-tion band valley degeneracy for Si into a doublet and a qua-druplet with the doublet;200 meV lower in energy. As aresult, each electron level is fourfold degenerate, two fromspin and two from valley. In the Ge sample, the biaxial com-pressive strain removes the valence band degeneracy at thezone center, leaving theu3/2,63/2& states;110 meV lowerin energy than theu3/2,61/2& states. In both cases, thestrain-split energy differences are large compared to otherrelevant energies, rendering the band structures simpler thanthose of the unstrained materials.

Each sample was mesa etched into a standard Hall barwith a channel width of 50mm and a distance of 400mm

a!Electronic mail: shsong@ee.princeton.edu FIG. 1. Schematic layer profiles for~a! the Ge sample and~b! the Si sample.

3422 Appl. Phys. Lett. 70 (25), 23 June 1997 0003-6951/97/70(25)/3422/3/$10.00 © 1997 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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between the voltage probes. Ohmic contacts were made byalloying AuAl for the Ge sample and AuSb for the Si samplein ambient forming gas. The standard four-terminal low fre-quency ac lock-in technique was used to measure the diago-nal resistances (Rxx). For the Ge sample, the density and themobility are p56.431011 cm22 and mp533,000 cm2/V s, respectively, atT50.3 K. For the Si sample, electronsare frozen out at low temperatures. However, it showed apersistent photoconductivity effect and, after illuminationwith a red LED for a short period of time, we were able toproduce a 2D electron gas withn55.231011 cm22 andmn

516 400 cm2/V s at T50.3 K. Two sets of low magneticfield (B) traces have been measured for each sample. Thefirst set was taken at variousT’s by passing a small fixedcurrent I ~10 nA! to avoid the heating of the carriers. Thesecond set was taken at a fixed substrate temperatureT50.3 K by passingI ’s larger than 10 nA.

In Fig. 2, we show the Shubnikov de-Haas~SdH! oscil-lations for the two samples. In order to simplify the analysis,the nonoscillatory background magnetoresistances are sub-tracted out. The top two panels are for the first sets taken atfive differentT’s and the bottom two for the second sets atfive different I ’s. The numbers at the minima of the SdHoscillations represent the filling factors,n5nh/eB, wherenis the density of the carriers. The minima in the oscillationsfor the Ge sample in Fig. 2~a! correspond to evenn ’s result-ing from its twofold spin degeneracy and, the minima for theSi sample in Fig. 2~b! correspond ton ’s in multiples of 4from the twofold spin and the twofold valley degeneracies.We obtained an effective mass of (0.0960.1)me for the Gesample and (0.2060.01)me for the Si sample by fitting theT dependence of the SdH oscillations.9 Hereme is the bareelectron mass. These values are in good agreement with thevalues reported in the literature.10,11 By comparing the top

and bottom panels, one can see that passing a large currentheats the carriers above the lattice temperatureTL ~0.3 K!. Ithas been reported that dc currents produce the same heatingresults as low frequency ac currents.12

In degenerate semiconductor systems where electron-electron scattering is more frequent than electron-phononscattering, the ‘‘electron temperature model’’13 is known towork well. The carrier distribution function can be assumedto be solely determined by an electron temperatureTe that ishigher thanTL . We used the amplitude of the SdH oscilla-tions as a thermometer to deduceTe of the heated carriersunder the assumption that the energy relaxation rate is inde-pendent of the magnetic field.3,14

In Fig. 3, we plot the carrierTe vs I at n515,16 for theGe sample and atn514, 16 for the Si sample. We obtainedthe following power law by fitting the slope of the datapoints:

Te;I a, a5H 0.5060.02, for the Ge sample

0.4060.02, for the Si sample. ~2!

This relation is valid throughout the SdH regime. We at-tribute this exponent difference between the Ge and Sisamples to the difference in phonon dimensionality in thesesamples.

In order to obtain the energy relaxation rate from theabove relation, we use the model proposed by Arai.15 For anelectron subject to an applied electric fieldE, the energyacquired by drifting a distance of inelastic mean free pathl e , is eEle , which will smear out the electron distributionfunction. The smeared distribution can be represented by theeffective electron temperatureTe . In steady state, this resultsin kTe;eEle for largeE’s. For a Fermi gas,l e5(Dte)

1/2,whereD and te are the diffusion constant and the energyrelaxation time, respectively. In theT range we studied,D is

FIG. 2. Oscillatory components ofRxx in Shubnikov de-Haas regime for~a!the Ge sample and~b! the Si sample. The data in the top panels were takenat variousT’s with I510 nA, the data in the bottom panels atT50.3 K withvarious I ’s. The numbers at the bottom of SdH oscillations in top panelsrepresent the filling factors. The arrows in the bottom panels indicate thefilling factors of the data shown in Fig. 3.

FIG. 3. Effective carrier temperatureTe vs I . Open symbols are for the Gesample atn515,16 and closed ones for the Si sample atn514,16. Theerrors are no larger than the size of the symbols. The straight lines show thefits with respective powers.

3423Appl. Phys. Lett., Vol. 70, No. 25, 23 June 1997 Song et al. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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T independent and the resistance remains Ohmic, since theT-independent elastic scattering is dominant in oursamples.16 Consequently,

Te;E2/21a;I 2/21a. ~3!

From Eqs.~2! and ~3!, a which reflects the dimension-ality of the relevant phonons can be obtained: 2610% for theGe sample and 3610% for the Si sample. As mentionedearlier, for the phonons to be confined to 2D, large acousticmismatch between the active layer and the cladding layers isnecessary. In addition, according to Snell’s law of refractionin acoustic waves, the sound velocities in the active layerneed to be smaller than those in the cladding layers to havetotal internal reflections that confine the phonons. Unfortu-nately, the sound velocities of the strained layers of Ge andSi are not available, to the best of our knowledge. Neverthe-less, the values of unstrained Ge and Si are known and canserve as good indicators regarding the possibility of totalinternal reflections. The longitudinal and transverse soundvelocities for @100# Ge are 4.963105 and 3.583105 cm/s,and for @100# Si, are 8.483105 and 5.873105 cm/s,respectively.17 Indeed, judging from the velocity values andthe sample structures, total internal reflections appear pos-sible in the Ge sample, but not in the Si sample. Althoughthe possibility of partial internal reflection cannot be ruledout, our results suggest that the phonons show broadly 2Dbehavior in the Ge sample and broadly 3D behavior in the Sisample.

Disorder which can affect theT exponent6 appears neg-ligible in our samples. A sample is considered disordered ifthe carrier mean free path is smaller than the inverse of thethermal phonon wave vectorq5kT/\s, ~i.e., q• l el,1),wheres is the sound velocity. In semiconductors, disordercan be very well characterized by the mobility of the carriers.For both samples above 0.5 K,q• l el.1, regardless ofwhether longitudinal or transverse modes are involved, if weuse the velocity values from unstrained Ge and Si. Therefore,disorder does not play an important role in our measure-ments.

In summary, we carried out carrier heating measure-ments with two samples, one with 2D holes in a strainedGe/Si0.4Ge0.6 quantum well and the other with 2D electronsin a strained Si/Si0.7Ge0.3 quantum well. We observed apower law behavior in the current dependence of the carriertemperature,Te;I a. We find a50.5060.02 for the Gesample and 0.460.02 for the Si sample. This exponent dif-ference is explained by the difference in phonon dimension-ality between the samples using a model proposed by Arai.15

The authors would like to thank Dr. M. A. Stroscio andProfessor H. P. Wei for helpful discussions. This work wassupported by the ARO and the NSF.

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11D. Monroe, Y. H. Xie, E. A. Fitzgerald, P. J. Silverman, and G. P. Wat-son, J. Vac. Sci. Technol. B11, 1731~1993!.

12H. P. Wei, L. W. Engel, and D. C. Tsui, Phys. Rev. B50, 14 609~1994!.13R. Stratton, Proc. R. Soc. London, Ser. A246, 406 ~1958!.14G. Bauer and H. Kahlert, Phys. Rev. B5, 566 ~1972!.15M. R. Arai, Appl. Phys. Lett.42, 906 ~1983!.16Y. H. Xie, E. A. Fitzgerald, D. Monroe, P. J. Silverman, and G. P. Wat-son, J. Appl. Phys.73, 8364~1993!.

17A. Dargys and J. Kundrotas,Handbook on Physical Properties of Ge, Si,GaAs and InP~Science and Encyclopedia Publishers, Vilnius, Lithuania,1994!.

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