Specific heat of disordered Xe films at low temperatures

VOLUME 39, NUMBER 4 PHYSICAL REVIEW B 1 FEBRUARY 1989 Specific heat of disordered Xe films at low temperatures N. Steinmetz, H. Menges, S. Dutzi, * and H. v. Lohneysen Physikalisches Institut der Uniuersitat Karlsruhe, D 7500-Karlsruhe, Federal Republic of Germany W. Goldacker Kernforschungszentrum Karlsruhe, We report measurements low temperatures Institut fur Techmsche Physik, D 7500 K-arlsruhe, (Received 20 May 1988) (=6 K). Federal Republic of Germany (7 of the specific heat C (0.08 K & T K) of Xe films condensed at Low-T x-ray dilfraction indicates that the films are crystalline but strongly disordered, with very small crystallites. The specific heat is very large compared to bulk Xe, and the excess specific heat can be attributed to surface modes and to tunneling states. Disordered solids exhibit a large variety of lowtemperature behavior. Often these features can be traced back to atomic degrees of freedom not available in perfect crystals. One of the major manifestations of atomic dynamics at low temperatures are the tunneling states in amorphous solids. ' These states have been found in insulating and metallic glasses, and also in crystalline materials. In the latter case, the extra degrees of freedom result, for instance, from the possibility of atomic or molecular rotations, such as in KBr~ „(CN)„. The tunneling states are observed, e.g. , by their roughly linear contribution to the specific heat and via their strong interaction with thermal or acoustic phonons. ' Rare-gas solids are among the simplest solids (except the lighter ones where quantum effects become important). Therefore, it appears interesting to look into the possibility of atomic tunneling states in disordered raregas solids. Also, atomic dynamics is often studied numerically using model systems with a Lennard-Jones (LJ) pobe retential, and rare gases can to a certain extent garded as realizations of such model systems. In addition, generation of amorphous solids by rapid solidification is often simulated using LJ potentials. In the present paper, we report on specific-heat measurements on disordered Xe films condensed at low temK). In order to determine the structure, peratures similarly prepared films were investigated with a lowtemperature x-ray diffractometer. The main results of our study are as follows: Well below 1 K, we observe a specific-heat contribution varying linearly with T, with a large coefficient ( 0.5 mJ/mol K ), hinting at the possibility of tunneling states. In addition, we find a very large enhancement of the vibrational specific heat (compared to bulk Xe) up to the highest measuring temperature (7 K). Our x-ray measurements indicate that the films are crystalline with a very small grain size. Hence, our results constitute an example of a simple crystalline system with a large number of atomic degrees of freedom. The specific-heat experiments were carried out between 0. 1 and 7 K in a dilution refrigerator and He cryostat. Xe (99.996%) was condensed with rates between 0. 1 and 5 nm/s onto a single-crystal quartz substrate. Background mbar. The substrate was in weak pressure was & 10 thermal contact with the cold part of the cryostat. Thus, — — (=6 = 39 its temperature could be kept constant during evaporation K. The heat capacity was and could be as low as Prior to each run, the measured with the ac method. heat capacity of the empty substrate (with heater and thermometer) was measured. The advantage of a quartz substrate is that it could be used to directly determine the mass and, thus, the specific heat of the sample without any during annealing. ambiguity due to, e.g. , sublimation Typically, the film mass was 0.5 mg, from which a thickness of 1 pm can be inferred. The films could be annealed up to =50 K before any sublimation occurred. hours. A Annealing time t, was typically several systematic check showed negligible time effects for =6 = (=4) &1 }1. The x-ray diffractometer (Cu Ka radiation) was operated using a flow cryostat with the minimum temperature of 5 K. Film preparation was done in the same way as discussed above. All films exhibited a structure compatible with fcc structure. The inset of Fig. 1 shows, as an example, the (111) reflection (always measured at 5 K) of Xe films with three different treatments. For films deposited at Td-6 K only the Xe (111) refiection was found, indicating a preferential film growth (close-packed planes parallel to the substrate). This texture remained after annealing up to 50 K. A film condensed at 50 K showed all refiections allowed by the fcc structure factor. Although the single Bragg refiection observed for the films with Td =6 K could also be interpreted as the (002) reAection of an hey structure, the systematic shift of the diffraction maximum towards lower angles and a strong narrowing with decreasing disorder renders this possibility Intermediate annealing unlikely. stages (not shown) confirm this behavior. In particular, no indication of coexistence of the two possible phases (fcc and hcp) was seen. The lattice constant for the disordered films (a 6. 11 A for T, =6 K) is actually smaller than for bulk Xe (ab =6. 131 A). This is in marked contrast to the macroscopic density deficit of 35% found, e.g. , by optical meaor adsorption of He and H2 (Ref. 6) for Xe surements films condensed below 10 K. Obviously, our results suggest a rather open structure of these films with many voids and internal surfaces. Our observation a & ag is in accord with a recent calculation of the properties of rare-gas sur- = = — 2838 1989 The American Physical Society REEF REp( R 39 2839 600— 00 6p QPP o E E E gp 200 0' p 0 30 20 5 $0 0 I I 1.0 2.0 FIG Il symbols . Speclfic heat of Xe films plotte yl s deposited at Z- T'2 &/Tvs line d pos'ted at Td "" 7 =50 K; (c) at T annealing =5o K. 13ark„ using the (co»culations s "ow th temperatures o our tern perature I ls range) th e mean dist ance betwe ensu face y so0 p etween 3 K en bolss d enote results s foor two diff'erent s. -0. Ie ~Hd o d 1 igure 1 fil ) Finally, sohd t 50 K. d ' '"ear contributipn tpo is seen. The )in ear specific heat ea of r the disor ered films which vanishes go~dually upoo annealin g~ m ust arise from xcitations with a co e abo~'e-mentioned t nty pf states and f o glasses, it is tern ™ptingto attrig model de@eloped ' a omic tunnelin mo d »te them to at w d.iscuss i re detail thee two extra cont I ij ll in the disprd ere d films . ( ) surface contrib butipns tp t"e tunnel tion var y inng roughly as T b elow 4 K, an ing contribution "p«portional to T (;) H armonic. vibrationons with a linea tion (for ex ™ " sur ace modes es 2 ~ t~mpe~a ig owever imen sion am «e realistic treatment must tC olid Ii ) Tli ( enhancemmentof Cover Cb n en C for films conden e additional s per ature. B =C —Cb can b d t 1 p This t 1. epen ence ig. ' su uionb su s. pon subsel ing at intermediate t hea 'fi . T in the heat of Xe fil ~s Plotted at C/T vs &2 Perature rangee 0 .3 -22 K, after eat treatment at various tern peratures. From t op to bottom: ' film e»porated at 6 K ( circles and annealed at lo K FIg. 2. Specific p si ed at specifi heat of bulk X dashed 0.p 3.0 T'(K') T (K~) h 10 -T, PV 6 E E 'LJ Figure 3 shows C/T vs T b fil s. A t these low tern p er two diff coefticient f l ). line, d' Th tersection with th ear in T, with i h tribution olK, is clearl ob "'t'd a't 6 K and a~~ealed h 1 di e extrapolated) Cb., ' ', in particular, p I 0. 1 0. 2 0. 3 T {K) Low-temperature s ec FIG. 3. L ihT '=6K X fi 1 l as guides to Solid re intended lines are int d S no BRIEF REPORTS of course incorporate the discreteness of the surface. Allan and de Wette' calculated C, for fcc solids where the atoms interact through an LJ potential by means of a numerical simulation. From this work, we can obtain C, for the Xe (100) surface using the commonly accepted values a=3. 156X10 ' J and a =3.95&&10 ' m for the This is compared to the measured exLJ parameters. cess specific heat b, C in Fig. 4 (the linear contribution is negligible in this plot). Very good agreement in the T dependence of d, C and C, is found. The only free parameter in this plot is the absolute magnitude of C, This is directly related to the ratio of the number of surface atoms to the total number of atoms N, /N which depends on the size of the crystallites. The magnitude of C, needed to match the data in Fig. 4 corresponds to N, /N =0. 17. Assuming, in a very simple model, a random packing of crystallites of roughly uniform spherical shape and size, a 64% is obtained' which is in good packing fraction of agreement with the above-mentioned measurements of the macroscopic density. Taking 15 nm as a typical diameter for the most disordered films, as inferred from the xray diffraction measurements (Fig. 1), we obtain N, /N =0.16. In view of the various assumptions made in this simple model, the agreement is surprisingly good. Above 6 K, the measured data deviate systematically from the calculated C, . This could be due to the neglect of anharmonic eff'ects or of dynamical displacements at the surface in the calculation. ' However, since these deviations occur at the limit of our measuring range, we do not attach too much significance to them. Also, in this simple interpretation of our data entirely in terms of additional surface modes, we have ignored possible changes in the Debye behavior of the disordered films, i.e., of the "interior" of the grains. In view of the inhomogeneous structure of the disordered films, an interpretation of the enhanced specific heat in terms of a phonon-fracton crossover' might be intriguing. However, such a crossover would lead to a positive curvature in C/T vs T at low T, followed by an inflection point at roughly the crossover temperature, in clear disagreement to what is observed (cf. Fig. 1). (ii) We now turn to the linear specific-heat contribution which is resolved only well below 1 K (Fig. 3). There are two diff'erent possibilities which might lead to atomic tunneling dynamics at these low temperatures. The first of these starts from the idea that double-well potentials can arise for atoms at grain boundaries. The distribution of grain-boundary angles in disordered Xe could provide a natural explanation for the distribution of tunneling barriers and distances necessary to account for the roughly constant energy density of states. Upon annealing, as the grain boundaries and voids gradually disappear and the macroscopic density of the films approaches the bulk density, the tunneling density of states is expected to decrease, as observed. The second possible origin of atomic tunneling states is related to the fcc-hcp competi'lion in the heavy rare-gas 39 I Xe ~ film ~ ~ " C) E Vy ~ ~ 7 0. 5— ~ ~ — 1 e ~ l' 1 ~ Pg 0 0 l I 2 4 T(K) FIG. 4. Excess specific heat AC C —Cb vs T for a Xe film Td=6 K. Dashed line indicates surface contribution C, with calculated after Ref. 10. solids. It is well known that the relative free-energy diff'erence between both phases of Xe is only a fraction of a percent. ' In fact, calculations for all reasonable twobody potentials show that the hcp phase should be slightly more stable than the fcc phase, contrary to what is observed. Hence, the stacking fault energy in the (ill) direction (i.e., the preferred growth direction in our disordered films) is very small. ' In addition, there is probably a high dislocation density' in our films. Together, these features suggest that small groups of atoms may be able to fcc-hcp undergo a dynamical collective rearrangement and vice versa. This picture is somewhat similar to the dynamical P-co transformation in, e.g. , Zr-Nb alloys. Indeed, in these materials evidence for tunneling states was found. ' At present, a definite conclusion which of the above interpretations is correct, cannot be drawn. We slightly favor the second alternative because in nanocrystalline Pd alloys' where the first possibility for tunneling states should also come to bear, no indication of an enhancement of the linear specific heat (by 0.5 mJ/molK ) over the electronic contribution was observed. In any case, the existence of low-energy excitations which are attributed to atomic tunneling states in disordered rare-gas crystals is well established by the present work. Recent ultrasonic measurements' have also shown evidence for tunneling states. In conclusion, our specific-heat results reveal that disordered rare-gas films exhibit a surprisingly rich variety of atomic dynamics at low temperatures. We hope that this work will stimulate molecular-dynamics calculations for double-well potentials in LJ solids. Our results are also of relevance in view of the widespread use of rare-gas solids for matrix isolation in cluster research. — BRIEF REPORTS Present address: ANT Nachrichtentechnik G.m. b. 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