Temperature dependence of the thermal expansion of AlNStephan Figge, Hanno Kröncke, Detlef Hommel, and Boris M. Epelbaum Citation: Applied Physics Letters 94, 101915 (2009); doi: 10.1063/1.3089568 View online: http://dx.doi.org/10.1063/1.3089568 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/94/10?ver=pdfcov Published by the AIP Publishing

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Temperature dependence of the thermal expansion of AlNStephan Figge,1,a� Hanno Kröncke,1 Detlef Hommel,1 and Boris M. Epelbaum2

1Institute of Solid State Physics, Section Semiconductor Epitaxy, University of Bremen, Otto-Hahn-Allee,D-28359 Bremen, Germany2Department Werkstoffwissenschaften, Friedrich-Alexander-University Erlangen-Nürnberg, Martensstraße 7,D-91058 Erlangen, Germany

�Received 26 August 2008; accepted 22 January 2009; published online 12 March 2009�

The thermal expansion of wurtzite AlN bulk crystals grown by physical vapor transport was studiedby high resolution x-ray diffraction in a temperature range from 20 to 1250 K. The temperaturedependence of the derived anisotropic thermal expansion coefficients along the a- and c-directionscould be well described over the entire temperature range within both the Debye model and theEinstein model. In comparison to GaN, larger expansion coefficients and higher characteristictemperatures have been found. The resulting thermal mismatch of AlGaN/GaN heterostructures arepresented. © 2009 American Institute of Physics. �DOI: 10.1063/1.3089568�

Due to the high band gap of 6.2 eV, AlN is a promisingmaterial for optoelectronic devices up to the mid-UV such asused for disinfection systems. Furthermore, it has a highthermal conductivity which is useful for high power and highfrequency devices. One major problem in such devices is thestrain of pseudomorphic grown ternary materials which leadsto bending and cracking of the structures. As the strain ispartially induced by the difference in thermal expansion co-efficients �TECs�, an exact knowledge of the parameters isneeded. While the anisotropic TECs of GaN and sapphire arewell known in a broad temperature range, the available TECsof AlN are based on powder diffraction and neglect the an-isotropy of TECs in c- and a-directions1 or originate fromthin heteroepitaxial layers and overestimate the anisotropy inthis case.2 Due to this fact, even at room temperature theabsolute values are varying in literature by more than 20%.Moreover, thermal expansion is a fundamental property ofthe crystal solid, as it is connected with the anharmonicity ofthe interatomic potential and the intrinsic energy. Therefore,the TECs show a strong correlation to the specific heat.

In the present study, precise data on the thermal expan-sion of AlN bulk crystals over a wide temperature range of12–1250 K are provided. Because the data can be accuratelydescribed by models for the phononic system, reliable valuesfor the characteristic temperatures of AlN are derived aswell.

The two AlN bulk crystals investigated within this studywere grown in tungsten crucibles using physical vapor trans-port method. The typical growth temperature of the crystalsis 2250 K during nucleation whereas the source region in thereactor is kept at 50 K at a higher temperature �for furtherinformation, see Ref. 3�. The first sample was growing in thenucleation area of the reactor. Due to several nucleation cen-ters in this reactor parts this sample has a multigrain �MG�appearance. The second crystal is the one taken from thereactor wall where it is spontaneously nucleated as a singlecrystal �SC�. The sizes of the samples are 4�4�2 mm3 forthe SC sample and 18�8�4 mm3.

Temperature dependent x-ray measurements were per-formed with a high-resolution diffractometer �Philips, X-Pert

MRD� equipped with a twofold hybrid monochromator and athreefold channel cut Ge�220� analyzer. The samples wereinvestigated in the temperature range from 20 to 400 K in anOxford cold head cryostat and from 300 up to 1025 K usingan Anton Paar HTK 1200 high temperature camera system�HTC�. For accurate determination of the temperature thethermocouple readings of both setups turned out to be insuf-ficient. Therefore, the temperature of the samples has beenmeasured utilizing a silicon reflector glued with a silver pasteto the top of the samples. The temperature was derived fromthe measured silicon �111� lattice parameters and literaturedata.4 This method yields above room temperature an accu-racy of �5 K. At lower temperatures the thermal expansioncoefficient of silicon drastically decreases. Therefore, thecold head temperature has been assumed to be accuratewithin broader limits of �12.5 K.

The AlN lattice parameter c was determined directly bymeasuring the scattering angles with the analyzer crystal5 forsymmetrical �002� and �006� reflections. The a-lattice con-stant was determined from the asymmetrical �105� and �205�reflections by taking the already determined c-lattice con-stant into account. In general the lattice plane spacings dhklhave been determined by taking refractive index correctionsproposed by Fewster and Andrew5 into account. In the caseof the MG-sample rocking curves the reciprocal space mapsrevealed a poly grain appearance and special care was takenin the temperature dependent measurements in probing thevery same crystal region.

At room temperature the lattice parameters c and a ofboth samples have been measured utilizing a standardEulerian cradle and are consistent with the literature datawithin the accuracy of the experimental setup �4�10−5 and1�10−4 Å, respectively�. Same results have been obtainedusing the cryostat. In case of the HTC special care had to betaken for the sample mounting. Due to the low window pro-file of 1 cm of the HTC a tilting in the �- or �-angle resultsin an intensity reduction of 20% per degree. This hinders theaccurate alignment of the sample at Bragg condition if thesample is not exactly coplanarly mounted. This effect is mostpronounced for low index reflexes. However, as this is asystematic error, a tilted mounting results into a constantdeviation in the lattice constant and does not affect the cal-culation of the TEC. Comparison of the data taken from thea�Electronic mail: figge@uni-bremen.de.

APPLIED PHYSICS LETTERS 94, 101915 �2009�

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�002� and �006� reflexes showed slightly different lattice con-stants but resulted in the same TECs. The MG-sample exhib-ited in the XRD-rocking curve with a full width at half maxi-mum �FWHM� of 0.15°, whereas the SC-sample showed amuch narrower FWHM of 0.005°. Therefore, the averageangle between the single grains in the MG-sample is in therange of a tenth degree. This angle is low enough not tocouple with the TECs of the different directions. Consequen-tially, both samples showed similar results in the determina-tion of anisotropic lattice constants and TECs, determinedfor the first time on bulk crystals.

In Fig. 1 the measured lattice constants a and c areshown in the temperature range from 20 to 1300 K of bothsamples in combination with the literature data.1,2,6–8 In bothlattice directions almost no thermal expansion is seen at lowtemperatures and an almost linear expansion at temperaturesabove 750 K. This behavior is in concordance with modelsbased on the intrinsic phonon energy of a solid system.Therefore, we fitted the data to models based on the Debye-and Einstein-like phonon dispersion. The anharmonicity ofthe interatomic potential is assumed to be only of the thirdpower in atomic distance and thus the lattice parameters aredirectly determined by the intrinsic energy.9 Therefore, thelattice parameter c can be described depending on the tem-perature T,

c�T� = c�0K� � �1 + �� � � � f��/T�� , �1�

where �� is the lattice expansion coefficient in the high tem-perature limit, � is the characteristic temperature, and f iseither the Debye or Einstein function defined as follows:

fD�x� = 3 � �0

1 t3

etx − 1dt , �2�

fE�x� =1

ex − 1. �3�

The Debye model is supposed to be more accurate forlow temperatures as only acoustic phonons are excited in thisrange. However, both models give a sufficient approximationof the transition range around the characteristic temperatureand result in linear expansion at high temperatures. In Fig. 1

the fitted Debye function is plotted for both lattice param-eters as a solid line. The Einstein function �dashed line�shows similar reproduction of the temperature dependent lat-tice parameter and can be only distinguished from the Debyemodel at low temperatures. Figure 2 shows the derived TECin the measured temperature range in comparison to the lit-erature data. In contrast to the data from Ivanov,6 no negativeTECs have been found at low temperatures. Therefore, thenegative TECs might be attributed to the ceramic samplesused by Ivanov6 which show a strong coupling of the aniso-tropic TECs.

The corresponding model parameters are shown togetherwith data from GaN in Table I. In comparison to GaN thecharacteristic temperatures are by a factor of 1.4 and 1.6higher in the c-direction and in the perpendicular direction,respectively. This ratio can be compared with the speed ofsound which is proportional to the Debye temperature.9 Fromelastic constant Cii and density � the speed of sound can becalculated as v=�Cii /�. Typical data from literature for theelastic constants11 yield a ratio of 1.4 in all crystallographicdirections. Therefore, the determined expansion parametersare in good agreement with other methods.

The effect of the different TECs onto heteroepitaxialstructures can be seen in Fig. 3. Here the temperature depen-dent lattice mismatch is plotted for AlN on GaN. The maxi-mum thermal mismatch between these two materials isaround 700 K. As the typical growth temperatures for ni-trides are at higher temperatures, the heterostructures thatsuffer during the cooldown stronger stresses than during

FIG. 1. �Color online� Temperature dependent lattice parameter of AlN inc-direction �upper part� and a-direction �lower part�. FIG. 2. �Color online� Temperature dependent TEC of AlN.

TABLE I. Model parameters determined by XRD in this work of AlN and ina former work of GaN �Ref. 10�.

Debye model Einstein model

��

�10−6 /K��

�K���

�10−6 /K��

�K�

�cAlN 5.8�0.1 1317�25 5.6�0.1 937�25GaN 5.73�0.44 898�24 5.71�0.43 662�18

�cAlN 7.1�0.3 1455�25 6.9�0.3 1025�25GaN 6.24�0.41 868�20 6.21�0.35 636�13

101915-2 Figge et al. Appl. Phys. Lett. 94, 101915 �2009�

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growth. Therefore, additional cracking of such structures canoccur upon cooldown.

The availability of bulk AlN crystals enables the reliabledetermination of TECs. In this paper the expansion in a tem-perature range from 20 K to 1250 K has been determined and

fitted with the Debye and Einstein models. The characteristictemperatures are in good agreement with other literaturedata. The comparison to data of GaN proposes a maximumin lattice mismatch at a temperature of 700 K.

This work has been partially funded within a researchgroup of the German Science Foundation �DFG� under Con-tract No. FI 1223/2-1.

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FIG. 3. �Color online� Thermal mismatch of AlN and GaN depending on thetemperature.

101915-3 Figge et al. Appl. Phys. Lett. 94, 101915 �2009�

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