Cross-plane thermal conductivity reduction of vertically uncorrelated Ge / Si quantum dot superlattices

A drastic reduction in temperature dependent cross-plane thermal conductivity κ⊥ occurs in Ge quantum dot superlattices (QDSLs), depending on the vertical correlation between dots. Measurements show at least a twofold decrease of κ⊥ in uncorrelated dot structures as compared to structures with the same Si spacer of 20nm but good vertical dot alignment. The observed impact of disorder on the conductivity provides an alternative route to reduce the thermal conductivity of QDSLs. The results of this work have implications for the development of highly efficient thermoelectric materials and on-chip nanocooling devices.

SiGe nanostructures are very promising for thermoelectric cooling of microelectronic devices and high-temperature thermoelectric power generation.][8][9][10] Previous studies by Lee et al. on short period 3 -7 nm Si/ Ge SLs unveiled a decrease of Ќ with decreasing SL period. 1 A minimum value of about 3 W / mK was measured at 300 K.For larger periods, L Ͼ 13 nm, they unexpectedly observed a decrease of Ќ which was attributed to the existence of extended defects.On the contrary, Borca-Tasciuc et al. 2 measured symmetrically strained Si 0.5 Ge 0.5 SLs with periods of 4 -14 nm without observing a clear dependence of Ќ on the SL period.Huxtable et al. 3 have shown the influence of the acoustic interface mismatch ͑AIM͒ in the thermal conductivity of Si/ Si 0.7 Ge 0.3 and Si 0.84 Ge 0.16 / Si 0.76 Ge 0.24 SLs.Further nanostructuring and phonon confinement may result in enhanced figures of merit and therefore Ge QDSLs have also been analyzed by a number of authors. 6-10Liu et al. 6,7 measured Ќ in Ge QDSLs with Ge content from 7% to 20% and a Si spacer of 20 nm.They reported values as low as 6.2 W / mK for samples grown at 500 °C with an average composition of Si 0.93 Ge 0.07 and dot surface coverage about 10%.Bao et al. 8 and Shamsa et al. 9 analyzed QDSLs grown by molecular beam epitaxy ͑MBE͒ at 550 °C with average Ge concentrations from 6% to 9% and a Si spacer of 20 nm and found values between 8 -12 W / mK at 300 K.More recently, Lee and Venkatasubramanian 10 measured Ќ in QDSLs grown by metal organic chemical vapor deposition at 750 °C and concluded that low values of Ќ can be attained either by using low SL periods or by increasing the dot coverage.They reported values between 2 -2.7 W / mK at short periods of 4 nm and areal densities around 20%.
All these studies deal with vertically correlated QDSLs.The vertical correlation between Ge dots can be modified by a proper selection of the spacer layer thickness.A larger thickness results in a lower correlation function.However, there is a serious drawback of this approach to increase the figure of merit in thermoelectric materials, since the electrical conductivity is expected to diminish with increasing Si spacer width.To circumvent this situation, one can use a seed layer of C to counteract stress memory between layers.Several authors [11][12][13] have shown that adding submonolayer amounts of C drastically affects the dot-nucleation mechanism, which can be successfully employed to grow uncorrelated nanostructures without an additional increase of the Si spacer.
In this letter, we report on the temperature-dependent thermal conductivity measured on Ge QDSLs grown simultaneously on the same Si wafer, which differ only in the degree of vertical correlation of the dots.We observe a decrease of the cross-plane conductivity for the uncorrelated samples in excess of a factor of 2. We interpret such a drastic impact of disorder on Ќ by using a theoretical approach based on a modification of the Fourier transport equation.
Samples were grown by solid-source MBE on Si ͑001͒ substrates as described elsewhere. 13By shuttering half of the wafer area in each deposition, it was possible to grow two multilayer structures on each Si wafer.In the area with C only in the seed layer, there is perfect vertical dot correlation between layers, whereas in the area with C deposited in each layer the correlation is almost completely suppressed.The growth temperature was fixed at 500 °C and the Si spacer at 20 nm for both samples.Table I shows the main parameters of sample 52.The vertical arrangement of the Ge dots in each case was confirmed by transmission electron microscopy and Raman scattering. 14The cross-plane thermal conductivity was measured with the differential 3 method in the temperature range 77-300 K. 15 A thin film of 10 nm Ti/ 100 nm Pt is deposited onto a 200 nm thick SiO 2 insulating layer previously grown on both the sample surface and a͒ Author to whom correspondence should be addressed.Electronic mail: javier.rodriguez@uab.es. the Si wafer used as reference.The thin film heater defined by photolithography and lift-off is 1 mm longϫ 15 m wide.
The thermal contribution of the SiO 2 / Si is measured independently and subtracted from the measurements on the SiO 2 / QDSL/ Si stack.The heater-width-to-film thickness and the film/substrate thermal conductivity ratios are 75 and about 0.05-0.1 at 300 K, respectively.Based on these values the heat spreading in the parallel direction of the film is very small, thus, the cross-plane thermal conductivity of the film calculated from the one-dimensional steady-state heat conduction model would be within 5% close to the true value for the film. 16Figure 1 illustrates the temperature rises in the correlated and uncorrelated QDSLs and the reference sample as a function of the modulation frequency.The input power was 35 mW at 300 K.The temperature drop across the uncorrelated film is constant at 0.090 K over a wide frequency interval.The thermal conductivity values derived for the undoped Si substrate and the 200 nm SiO 2 film yield 150 W / mK and 1.36 W / mK at 300 K, respectively, which closely agree with literature values. 15,17,18igure 2͑a͒ shows the thermal conductivity as a function of temperature for sample 52 which is representative of a series of Ge QDSLs, being this particular one also studied by Raman scattering.The equivalent Ge fraction of the QDSL is estimated to be 3%.The thermal conductivity of an undoped Si 0.97 Ge 0.03 alloy at 300 K is about 17 W / mK, 19 slightly above the value measured for the correlated sample, 14.5 W / mK.This value can be qualitatively compared to results of Refs.8 and 9 which obtained 12 W / mK at 300 K for a QDSL of overall composition Si 0.94 Ge 0.06 grown by MBE at 550 °C.][8][9][10] The Ge content and the total surface coverage of the dots in our QDSLs ͑see Table I͒ are below those reported in earlier studies rendering difficult a direct comparison.If we extrapolate data from Lee et al. 10 to 4%-5% coverage and 20 nm Si period, a value about 20 W / mK may be inferred.The lower value that we obtain may be due to the lower growth temperature which favors a high Ge content within the dot and therefore increases the AIM of the SL.At low temperatures, the crossplane thermal conductivity increases monotonically with temperature remaining then at a relatively constant value between 200 and 300 K.This behavior is in frank contrast to that of bulk Si which exhibits maximum conductivity at 40 K but decreases to a value of ϳ150 W / mK at 300 K.The striking result of this work is the further reduction of the room-temperature thermal conductivity by a factor in excess of 2, which is attained just by destroying the vertical correlation between dots.In this respect, it is very instructive to compare the results of thermal transport with those from Raman scattering.As illustrated in Fig. 2͑b͒, in QDSL structures, there is a close relationship between the magnitude of the thermal conductivity and the spectral features of light scattering by acoustic phonons. 14For the multilayer sample with perfect dot correlation in the growth direction, the interaction of the acoustic phonons with the ensemble of electronic states confined to the dots gives rise to well-defined Raman interferences.The interference contrast almost vanishes when carbon is introduced on the dot-nucleation surfaces in each layer.Instead, a strong and decreasing background is observed at small Raman shifts ͓bottom spectrum in Fig. 2͑b͔͒.These drastic changes in the Raman spectra of dot multilayers with and without carbon are directly related to the existence or lack of vertical correlation between Ge dots, as confirmed by calculations within the Raman interference model reported in Ref. 14.
To understand the observed thermal conductivity reduction, we compute Ќ by an approach based on an extension of the Fourier heat transfer equation that yields an analytical expression for the thermal conductivity of nanosystems of given size, 20 Si,Ge = 0 L 2 where 0 is the material bulk thermal conductivity, ᐉ is the mean phonon free path, 21 and L is the size of the system in the direction of propagation.The ordered ͑disordered͒ sample is formed by QDs which are vertically correlated ͑uncorrelated͒, as schematically illustrated in the inset to Fig. 2͑a͒.Using an average dot base diameter of 30-40 nm and dot densities about 5 ϫ 10 9 cm −2 ͑see Table I͒, the surface fraction covered with dots f dots , in the eight-period stacks of the ͑un͒correlated samples amounts 0.5 and 0.05, respectively.The effective thermal conductivity is written as the weighted sum of two parallel thermal processes, where B1 represents the thermal conductivity contribution due to the presence of QD ͑region B1͒ and B2 is the thermal conductivity contribution in the regions without Ge dots ͑region B2͒.B2 is the dominant thermal path in the correlated sample.To model Ќ we use the two elementary building blocks of Fig. 2͑a͒.Block 1 is formed by a dot of height L d and a silicon spacer of width L s1 .The resistance is given by the sum of four series resistors, i.e., the resistance of the dot r d , the resistance of the spacer r s1 , and two interface thermal resistances r i ͑Kapitza resistances͒, due to the difference of acoustic properties and interface quality at the dot/matrix interface.The resistance of the block is not dependent on the height of the dot due to the linearity of Eq. ͑1͒ in the low size regime.Therefore, the variation of the Ge dot height plays a minor role on the effective thermal conductivity compared to disorder.Block 2 is mainly composed of Si, with a width L s2 = L s1 + L w .and the resistance is the sum of the resistance of the silicon spacer r s2 and the interface thermal resistance imposed by the wetting layer r w .In all cases, the thermal conductivity is readily obtained by substituting the lengths and mean free paths in Eq. ͑1͒.We use Eq.͑2͒ to calculate the Ќ across the sample ͓solid lines in Fig. 2͑a͔͒.The thermal boundary resistances ͑TBRs͒ at 300 K that provide a good fit are r i =3ϫ 10 −8 m 2 K / W and r w = 2.5 ϫ 10 −11 m 2 W / K for the dot/matrix interface and the wetting layer, respectively.We have assumed that the temperature dependence of the TBR follows predictions of the diffusive mismatch model ͑DMM͒. 22The value obtained at 300 K for the dot/matrix resistance is of the same order of magnitude as previously reported for other systems 23,24 and larger by an order of magnitude to the value of 3 ϫ 10 −9 W / m 2 K derived from DMM at 300 K.However, the value associated with the thermal resistance of the wetting layer is significantly smaller probably due to phonon tunneling effects across it.Despite its simplicity, the agreement between the model based on Eqs.͑1͒ and ͑2͒ and the experimental data reinforces the concept that the reduction of thermal conductivity between correlated and uncorrelated samples is entirely due to the strong randomization of the dot spatial distribution induced by the presence of C in every layer.It is expected that lower thermal conductivities could be attained by doping and by reducing the Si spacer thickness which leads to an effective increase of the number of interfaces per unit length in the growth direction.Therefore, we anticipate that a combination of a short period with a random stacking of the dots in the vertical direction may lead to Ge QDSLs with even smaller values of Ќ , which in combination with a high electron mobility Si-rich nanostructures may lead to unprecedentedly large figures of merit.
In conclusion, we have shown that at room-temperature uncorrelated Ge QDSLs with a thin Si spacer of 20 nm exhibit at least a twofold decrease in thermal conductivity, as compared to ordered structures with the same spacer thickness.A model that incorporates the microstructure of the sample shows very satisfactory agreement with the experiment.This work provides an alternative route to reduce the cross-plane thermal conductivity.Thus, our results are important for the development of a different strategy to attain improved figures of merit for thermoelectric applications of nanostructured materials based on stacks of Si/ Ge QDs.

FIG. 1 .
FIG. 1. Temperature rise as a function of modulated frequency at 300 K for the correlated and uncorrelated 160 nm Ge QDSL and the reference sample.

TABLE I .
Main characteristics of the MBE samples.