Depth-dependent critical behavior in V2H
Charo I. Del Genio, Johann Trenkler, Kevin E. Bassler, Peter Wochner, Dean R. Haeffner, George F. Reiter, Jianming Bai, Simon C. Moss
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Depth-dependent critical behavior in V H Charo I. Del Genio,
1, 2
Johann Trenkler,
1, 3, ∗ Kevin E. Bassler,
1, 2
Peter Wochner, Dean R. Haeffner, George F. Reiter, Jianming Bai, and Simon C. Moss
1, 2 University of Houston, Department of Physics, 617 Science & Research 1, 4800 Calhoun Rd, Houston, TX 77204-5005 Texas Center for Superconductivity, Houston, TX 77204 Max Planck Institut f¨ur Metallforschung, D-70569 Stuttgart, Germany Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439-4815 Oak Ridge National Laboratory, Oak Ridge, TN 37831 (Dated: November 2, 2018)Using X-ray diffuse scattering, we investigate the critical behavior of an order-disorder phasetransition in a defective “skin-layer” of V H. In the skin-layer, there exist walls of dislocation linesoriented normal to the surface. The density of dislocation lines within a wall decreases continuouslywith depth. We find that, because of this inhomogeneous distribution of defects, the transitioneffectively occurs at a depth-dependent local critical temperature. A depth-dependent scaling lawis proposed to describe the corresponding critical ordering behavior.
PACS numbers: 61.05.cp, 64.60.Cn, 64.60.F-, 64.60.Kw, 61.72.Dd
I. INTRODUCTION
Structural defects exist in almost all real crystallinesolids. Therefore, in order to understand structuralphase transitions, it is crucial to understand the influ-ence that defects can have on ordering behavior. It hasbeen shown that defects, through their accompanyingstrain fields, can change the nature of phase transitions,including their universal critical properties . Defectshave also been shown to be responsible for the appear-ance of the so-called “central peak” in diffuse X-ray orneutron scattering and the related “two-length scale”phenomena observed in an number of experimental sys-tems . These previous studies haveimplicitly assumed that defects are homogeneously dis-tributed in the material, at least in the region of thecrystal being studied, even if that region is only a “skin-layer”, i.e., a near-surface region in which the defect den-sity is known to be different than in the bulk. In realsystems, however, defects are often caused by surfacetreatments. In this case, they can be inhomogeneouslydistributed, occurring mostly in a skin-layer with a den-sity that continuously decays into the bulk over severalmicrons . Such an inhomogeneous distribution ofdefects complicates the ordering behavior of many realcrystals. In this letter, using diffuse X-ray scattering inboth reflection and transmission geometries, we analyzethe depth-dependent critical behavior of the structuralordering of a crystal with this type of inhomogeneousdefect distribution.Divanadium hydride (V H) is an interstitial alloy thatundergoes a structural phase transition between an or-dered monoclinic phase β and a disordered body cen-tered tetragonal phase β as temperature is increased (forthe phase diagram see Ref. 16). In the crystal we study,defects occur almost exclusively in a skin-layer that ex-tends several µ m below the surface. In the skin-layer,there exist walls of dislocation lines oriented normal tothe surface . The density of dislocation lines in the walls decreases with depth. As we will see, the character ofthe structural transition in the crystal can change radi-cally with the depth at which it takes place. In the bulkmaterial it is first order , but it becomes continuous inthe skin-layer and has a critical temperature and criticalproperties that depend on depth. We propose a modifi-cation of the scaling law for the inverse correlation lengththat accounts for its depth-dependence and allows us totreat the scattering measurements at different depths ina unified framework. II. EXPERIMENTAL SETUP AND RESULTS
For our experiments we used a thin plate (0.96 mm)of a vanadium single crystal loaded with purified hy-drogen, so that it had a bulk concentration ratio of c H /c V = 0 . ± . α X-rays at a rotating anode source and atseveral energies on X14A at the NSLS at BrookhavenNational Laboratory. The high energy transmission ex-periments were carried out with 44.1 keV X-rays at theundulator beamline SRI-CAT, 1-ID, at the APS at Ar-gonne National Laboratory . In all the experiments weconfirmed the absence of higher harmonic contamination.Since we earlier observed two length scales in this crys-tal we used the different scattering geometries and en-ergy ranges in order to detect separately the influencesarising from the bulk and the skin layer. In all cases thesample was mounted in a strain-free manner in a vac-uum of ∼ − torr. The temperature fluctuations of theentire setup were less than 0.05 K at T >
443 K.We have four indications of a defective near-surfacelayer on our sample: (1) A hydrogen and oxygen gradi-ent measured by high resolution elastic recoil detectionanalysis (HERDA) in the first 150 ˚A ; (2) an oxygen gra- (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) C (cid:8) (cid:10)(cid:10)(cid:11)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:12)(cid:10)(cid:10)(cid:13)(cid:10)(cid:10)(cid:14) (cid:1) (cid:15)C4 µ (cid:16)31 21 (cid:19)1 (cid:11)1 (cid:10)1 (cid:12)1 (cid:13)1 (cid:1) FIG. 1: Critical temperature T extC versus depth d . dient measured by secondary neutral mass spectroscopy(SNMS) in the first 150–200 ˚A ; (3) the decay of the mo-saic spread with depth ; (4) a larger d -spacing in thenear-surface region than in the bulk . HERDA showsthat the hydrogen content increases with depth until thebulk concentration is reached at a depth of about 150 ˚A.In our most surface sensitive experiment we have used9 keV X-rays and low momentum transfers to measurethe influence of the upper 150 ˚A on the scattering; thiscontribution appears to be about 0.2 %, as deduced fromthe fraction R of the intensity diffracted down to a depth d in the sample. In a symmetric scattering geometry, R is given by R = 1 − e − µd sin Θ , (1)where µ is the linear absorption coefficient of X-rays andΘ is the Bragg angle . The only defects penetrating upto several microns in our sample are thus the walls ofdislocations responsible for the mosaic spread. We notethat there is a depth-dependent stress field associatedwith this decay, caused by walls of dislocation lines .We focus here on the depth dependence of the crit-ical behavior in the vicinity of the β - β order-disordertransition in the several micron thick skin-layer. The cor-relation length ξ = 1 /κ for T > T extC at a constant depthis the inverse of the half width at half maximum of thecritical diffuse scattering (CDS) profiles, and could bereliably determined only through a fit involving a convo-lution of the measured resolution function and the CDSprofiles. The critical temperature T extC ( d ), whose valuedepends on the depth d (see Fig. 1), is defined here asthe extrapolated temperature at which the full width athalf maximum reaches 0. Note that T extC ( d ) is an “inte-grated” quantitity that depends on the scattering fromthe entire skin-layer down to depth d . The fact that T extC ( d ) depends on depth, presumably implies that thereis a real, local critical temperature ˆ T C ( d ) that also de-pends on depth. Note also that the concept of a local κ ( (cid:3) (cid:4) (cid:5) t (cid:5)(cid:7) (cid:4)(cid:8) (cid:5)(cid:7) (cid:4)(cid:9) (cid:5)(cid:7) (cid:4)(cid:10) (cid:5)(cid:7) (cid:4)(cid:5) (cid:1) (cid:11)(cid:5)(cid:7) (cid:4)(cid:9) (cid:5)(cid:7) (cid:4)(cid:10) (cid:1) FIG. 2: (Color online) Inverse correlation length κ versusreduced temperature t = TT extC − µ m, the red squares to 13.1 µ m,the blue diamonds to 18.4 µ m, the pink upward triangles to25 µ m, the green downward triangles to 34 µ m and the orangestars to the bulk. critical temperature is well-defined only within a regionsmaller than the local correlation length. In other words,for any particular depth d ∗ , when the temperature ap-proaches T C ( d ∗ ) the ordering process will happen in alayer around that depth not thicker than the local corre-lation length.The high energy transmission experiments indicated astrong first order phase transition in the bulk, evidencedby a strong drop of the (0 5 / /
2) superstructure in-tensity by a factor of more than 400 at T extC (see Ref. 17).This is associated with a transition width of ≈ . β canbe determined from the integrated Bragg intensities ofsuperstructure reflections I : I ∝ Φ = − B (cid:18) TT extC − (cid:19) β , (2)where Φ is the Bragg-Williams order parameter, and B a constant. From the corrected intensities for the(0 5 / /
2) and (0 7 / /
2) superstructure reflec-tions, after excluding a small two-phase region, we ob-tained a value of β = 0 . ± .
02 (see Ref. 18), treating T extC as a fit parameter as, e.g., in Ref. 20.Since κ scales generally as κ = κ t ν , where here t = TT extC − , we estimatedthe correlation length exponent ν from the slope of a dou-ble logarithmic plot of the fitted κ versus t (Fig. 2); thevalues obtained were 0 . ± . ν . ± .
07 whenneglecting the crossover in ν and in the susceptibility ex-ponent γ at larger reduced temperatures . The range in c ( Å - ) (cid:1) d ( µ m)0 10 20 30 40 (cid:1) M - d ( µ m)0 10 20 30 40 FIG. 3: (Color online) Proportionality factor c (eq. 3) vs.depth d . The solid red line is the value of c calculated accord-ing to eq. 8. Inset: inverse of the mosaic spread (dimension-less) versus depth d . ν is essentially attributable to experimental uncertain-ties, e.g., the limited resolution for smaller t . Note thatwe measure essentially the same value of ν and of theother critical exponents regardless of depth. Thus, thevalue of the critical exponents are depth-independent .Our values of β , ν and γ (0 . ± .
13 as reported inRef. 14) all support tricritical behavior in the skin layerfor small t when the tricritical point is approached alongthe T -axis. Although the correlation length decreaseswith depth, our values of ν and γ (the subscript “1”refers to small t ) compare quite well with the theoreticalvalues of ν = 0 . γ = 1 obtained from the analysisof a metamagnet which yields mean field exponents .They also compare well with other systems presumed tobe tricritical, e.g., ν = 0 . ± .
08 and γ = 1 . ± . Cl . III. ANALYSIS AND THEORETICAL MODEL
According to the criteria defined by Krivoglaz , thecomposition of the sample was close enough to the tricrit-ical point to observe tricritical behavior. Although theexperimental value of β is smaller than the theoreticallyexpected value of 0.25, it is comparable with an earliermeasurement and with other tricritical systems if the in-fluence of the two-phase region is neglected . Giventhis good agreement between the values for the criticalexponent, we believe that tricritical behavior is dominantfor small reduced temperatures.Despite the fact that the critical exponent ν is depth-independent, there is a depth dependence to the mea-sured behavior of the correlation length, or, equivalently,to the inverse correlation length κ . To account for thisdepth-dependence, we propose that the scaling law for κ be modified such that κ ( d ) = c ( d ) t ν , (3)where the factor c is dependent on the depth d . To de-termine the function c ( d ) we calculated the y -interceptsof the fitted κ vs. t plots shown in Fig. 2. The resultingmeasured values for c ( d ) are shown in Fig. 3.These results indicate that, for the same change in t ν ,the inverse correlation length decreases faster for smallerdepths. On the other hand, the critical ordering beingmeasured is presumably occurring near the defect linesdue to their strain fields , which cause the appearance ofordered regions. We can then safely assume that the den-sity of ordered regions at any given depth is proportionalto the density of dislocation lines at the same depth. Themosaic spread gives us a measure of this density, so weexpect c ( d ) to be proportional to the inverse of the mo-saic spread. However, it should be noted that since κ ( d )is calculated from the half width of the CDS profiles atdepth d , it is an averaged measure. In fact, the CDSprofiles measure ordering throughout the skin-layer tothe depth that is probed. Thus, arguably, c ( d ) shouldbe proportional to an integral average of the inverse ofthe mosaic spread M from the surface to depth d : c ( d ) ∝ Z Z d M ( z ) d z . (4)The factor Z has to account for the absorption of theX-rays along the path through the material, so that it is,in itself, an integrated quantity: Z ( d ) = Z l ( d )0 e − µx d x , (5)where µ is the X-ray absorption coefficient and l is theeffective path into the material, which, for a depth d anda scattering angle ϑ is given by l ( d ) = 2 d sin ϑ . (6)We can then express Z directly as Z ( d ) = 1 µ (cid:16) − e − µd/ sin ϑ (cid:17) (7)and, absorbing µ into the proportionality, give our finalexpression for the experimentally measured c as c ( d ) ∝ R d M ( z ) d z − e − µd/ sin ϑ . (8)This quantity, calculated with average values of µ andsin ϑ and using a sigmoid function fit for M , is shown asa line in the main part of Fig. 3. The proportionality con-stant used, dependent on the particular sample and thedetails of the experiment, had a fitted value of 0 . − .In the inset of the same figure we show the depth depen-dence of the inverse of the mosaic spread. Notice also κ ' t ν FIG. 4: (Color online) Inverse correlation scaling function κ ′ versus scaled reduced temperature t ν , showing the datacollapse. The black circles correspond to 1.6 µ m, the redsquares correspond to 13.1 µ m, the blue diamonds to 18.4 µ m,the pink upward triangles to 25 µ m and the green downwardtriangles to 34 µ m. that in this estimate for the experimentally measured c , µ and sin ϑ only play a significant role for small depths,due to their presence in a negative exponential.With this treatment for c ( d ), it is possible to collapsethe scattering data for κ onto a single scaling function κ ′ = κc ( d ) (9)as shown in Fig. 4. Note that when t ν goes to 0 thecollapsed curves extrapolate to 0. This corresponds toa vanishing inverse correlation length, or, equivalently,to a diverging correlation length and thus suggests that T extC ( d ) is the actual critical temperature, or at least thatit is not too far from the real critical temperature ˆ T C ( d ).The tails of the curves, for t ν & .
07 correspond to thecritical region above the two-length scale crossover .The collapse is strong evidence that the density of de-fects directly drives the critical behavior in the material.The mechanism involved is very likely to be the inter-action with the stress and strain fields induced by thedefects, which is most probably also responsible for thecrossover to different values of the critical exponents andfor the change in the order of the transition between bulkand skin-layer. IV. CONCLUSIONS
In summary, we have observed a change in the order ofthe phase transition in a V H crystal, which is first-orderin the pure bulk and continuous in a skin layer that isseveral microns thick. Defects, in the form of dislocationlines, exist in the skin layer and are responsible for thechange in the ordering behavior. The density of thosedefects is inhomogeneously distributed and decays con-tinuously with depth from the surface. Throughout theskin layer, near the critical temperature, the measuredvalues of the critical exponents are those of a tricriticalpoint. Although the values of the critical exponents de-scribing the scaling behavior of thermodynamic functionsin the skin layer do not depend on depth, the coefficientsof the power law function of κ and, presumably, of otherthermodynamic functions, do show a depth dependence.Focusing on the behavior of κ , we find that its depth-dependent scaling coefficient depends on an integral av-erage of the inverse defect density. Using this fact andpostulating a modified scaling law for the inverse corre-lation length, we are able to collapse the measured dataonto a single scaling function. Thus, we are able to treatscattering measurements at different depths in a singleunified framework. Acknowledgments
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