Emergence of magnetism in bulk amorphous palladium
Isaías Rodríguez, Renela M. Valladares, David Hinojosa-Romero, Alexander Valladares, Ariel A. Valladares
EEmergence of Magnetism in Bulk Amorphous Palladium
Isa´ıas Rodr´ıguez, Renela M. Valladares, David Hinojosa-Romero, Alexander Valladares, and Ariel A. Valladares ∗ Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 70-542, Ciudad Universitaria, CDMX, 04510, M´exico Instituto de Investigaciones en Materiales, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 70-360, Ciudad Universitaria, CDMX, 04510, M´exico (Dated: July 11, 2019)Magnetism in palladium has been the subject of much work and speculation. Bulk crystallinepalladium is paramagnetic with a high magnetic susceptibility. Palladium under pressure and pal-ladium nanoclusters have generated interest to scrutinize its magnetic properties. Here we reportanother possibility: Palladium may become an itinerant ferromagnet in the amorphous bulk phaseat atmospheric pressure. Atomic palladium is a d element, whereas bulk crystalline Pd is ad − x (sp) x material; this, together with the possible presence of unsaturated bonds in amorphousmaterials, may explain the remnant magnetism reported herein. This work presents and discussesmagnetic effects in bulk amorphous palladium. I. INTRODUCTION
Atomic palladium, being the last d element in the 4throw in the periodic chart of the elements, displays a va-lence that may be a function of the molecule, compoundor of the dimensionality of the structure in which it par-ticipates. As a free atom it exhibits the electronic con-figuration of a noble gas but having the 4d shell filledand the 5s5p shells energetically accessible has led sev-eral authors to propose that palladium clusters and bulkpalladium under pressure could become magnetic. Bulkpalladium in its crystalline form and at atmospheric pres-sure is paramagnetic with a high magnetic susceptibility[1]. First principles simulations of palladium under neg-ative pressure [2] and experimental work in palladiumnanoclusters [3] indicate that these samples may displaymagnetic properties. Also, the fact that it has a high pa-rameter of Stoner [4] has made palladium a very appeal-ing subject. Calculations come and go, and results are re-ported, but experiment should say the final word. Couldit be then that amorphous palladium ( a -Pd) may alsodisplay interesting magnetic properties that would shedlight on a better understanding of magnetism in bulkmaterials, both defective and crystalline? The proper-ties of the amorphous phase have been little studied andconsequently this is terra ignota that must be explored. ∗ Corresponding author; e-mail: [email protected]
Motivated by these considerations, in this work we in-vestigate the effect of topological disorder in the elec-tronic and magnetic properties of amorphous samples ofbulk palladium at zero kelvin. We propose that atomicdisorder in solid palladium could generate magnetismsince this disorder would induce an unbalance in the num-ber of nearest neighbours, locally creating unsaturatedbonds leading to a net spin and consequently to a netmagnetic moment. This, together with the appearanceof holes in the corresponding d band, due to the spillingover of electrons unto the s and p bands may contributeto magnetism. We have performed ab initio calculationsand the results indicate that, in fact, magnetism mayappear in a -Pd: Palladium may become an itinerant fer-romagnet in the amorphous bulk phase at atmosphericpressure and at T = 0 K. Since our results are obtainedfor zero temperature, it is reasonable to ask: could itbe possible to find this magnetism for non-zero tempera-ture?. We believe the answer is yes, as long as palladiumis maintained at very low temperatures in an amorphousstate, in a similar manner as superconducting amorphousbismuth exists at T (cid:46) undermelt-quench approach. This approach allows the generation ofdisorder in an otherwise unstable crystalline structure, a r X i v : . [ c ond - m a t . s t r- e l ] J u l isodense to the stable one, by heating it to just belowthe melting temperature of the real material and thencooling it down to the lowest temperature possible. Inthis manner, a disordered specimen is created and thenan optimization run is carried out to release stresses andlet the sample reach local equilibrium. Our previouscomputational studies give us confidence in our proce-dure and therefore in our present results. However, seeFig. 1 in Ref. [8] where an amorphous phase analogousto the ones we obtained is reported and also comparewith the experimental Pd-Pd partial Pair DistributionFunction (pPDF) taken from Ref [9, 10] invoked later on. FIG. 1. Topology of amorphous palladium; the initially un-stable, diamond-like, supercell used contains 216 atoms. (a)Sphere representation of the three atomic structures compu-tationally generated. (b) Pair distribution functions for thethree ab initio simulated supercells shown in (a). The insetdepicts the structure of the bimodal second peak.
A variation of this approach consists in doing molec-ular dynamics also on an unstable specimen at a givenconstant temperature, under or over the melting point ofthe real material. The optimization (relaxation) run thenensues. For Pd we did precisely this on a supercell with216 atoms, and generated, using ab initio techniques,three amorphous structures presented in Fig. 1(a) withatoms represented by spheres and whose Pair Distribu-tion Functions (PDFs or g ( r )) are shown in Fig. 1(b).The bimodal structure of the second peak, typical ofamorphous metallic elements, can be seen in the inset.See also Ref. [8]. II. METHOD
The computational tools utilized are contained in thesuite of codes Materials Studio (MS) [11]. In particular,to perform the Molecular Dynamics (MD) and the Geom-etry Optimization (GO), and to calculate the electronicand magnetic properties of a -Pd, the code CASTEP wasused [12].A crystalline palladium supercell of 216 atoms was con-structed with diamond symmetry (unstable) and with theexperimental crystalline density of 12.0 g/cm ; this insta-bility allowed the undermelt-quench process to generateamorphous supercells, as mentioned in refs [5–7]. Thecell underwent three independent MD processes. Oncethey were complete the three resulting structures weresubjected each to a GO procedure starting with a totalspin of 93, 96 and 97 µ B generated by the MD on eachcell. The results indicate that the final topological struc-tures, determined through the PDF, are essentially thesame, Fig. 1(b). The energy and the Average MagneticMoment (AMM), in Bohr magnetons µ B , are shown inFig. 2.For the NVT MD the following approximations wereused. The PBEsol functional with a zero spin initially;an electron energy convergence tolerance of 2 x 10 − eVwith a convergence window of 3 consecutive steps; a cut-off energy of 260 eV to generate the plane-wave basis torepresent the 2160 electrons (10 per atom) distributedin 1297 bands (217 empty); a process at 1,500 K usinga thermal bath controlled by a Nose-Hoover thermostat,with a time step of 5 fs during 300 steps, for a total dura-tion of 1.5 ps, and a Pulay mixing scheme. To optimizethe MD process, the palladium ultrasoft pseudopotential,Pd 00PBE.usp, included in the MS suite of codes was thechoice, Figs. 2(a) and 2(b).For the GO the following parameters were employed.The minimization of the energy of the structure was per-formed with the density mixing method under the Pulayscheme; the functional PBEsol and the relativistic treat-ment according to Koelling-Harmon included in MS; theenergy cutoff for the plane waves was set to 300 eV; the2160 electrons (10 per atom) were distributed in 1353bands (226 empty). The initial spin was the output ofthe MD results: 93, 96 and 97 µ B ; an electron energyconvergence tolerance of 1 x 10 − eV with a convergencewindow of 2 consecutive steps and a smearing of 0.1 eV;the geometry energy tolerance used was 1 x 10 − eV; theforce tolerance used was 3 x 10 − eV/; the displacementtolerance used was 2 x 10 − ˚A and the geometry stresstolerance was set to 5 x 10 − GPa. Here the maximumnumber of steps was set to 1,000 to make sure it wouldrelax within the tolerance limits. The energy systemat-ically diminishes until an arrangement of atoms in localequilibrium is reached, Figs. 2(c) and 2(d).To investigate the magnetic properties, we ran boththe MD and GO processes with unrestricted spin, so themagnetism would evolve freely and acquire a value con-gruent with a minimum energy structure. The magnetic
FIG. 2. Molecular dynamics and geometry optimizations for the three palladium supercells. (a) Energy per atom as a functionof steps of MD. (b) AMM per atom (in Bohr magnetons µ B ) as a function of the MD steps. (c) Energy per atom as a functionof the GO steps. (d) AMM per atom (in Bohr magnetons µ B ) per step of GO (tends to 0.45 µ B per atom). The inset detailsthe behaviour in the first 100 steps. moment per atom begins to manifest in the first 50 stepsof MD and increases until the end of the run, Fig. 2(b).Afterwards, it increases somewhat during the GO pro-cess and tends to a constant value, 0.45 µ B per atom,Fig. 2(d). The inset shows details of the first 100 stepsof GO.How can we be sure that the PDFs obtained do repre-sent the amorphous structure of bulk palladium and thattherefore the AMM obtained corresponds to the amor-phous phase? We could argue that since our previousresults [5–7] are very close to the experimental ones, thePDFs that we report in this work should be adequate todescribe a -Pd; however, since to our knowledge nobodyhas experimentally produced the pure amorphous phase,we decided to validate our topological findings by usingsome experiments reported in the literature. Experimen-talists have obtained PDFs for amorphous palladium-silicon alloys: Masumoto and coworkers [9] studied thesealloys and determined a Pd-Pd pPDF, reported in ref [10] and reproduced in Fig. 3(b) in agreement with our simu-lations. Fukunaga et al . [13] studied a -Pd Si , whereasAndonov and collaborators [14] studied a -Pd . Si . .The system a -Pd Si was studied by Louzguine [15];all of them are shown in Fig. 3. We first compare thetotal PDFs they report for the alloys with our simulatedPDF for the pure, Fig. 3(a); where the similarities canbe observed. We next compare, in Fig. 3(b), our simu-lation with the experimental result by Masumoto et al .[9, 10] for a Pd-Pd pPDF; the agreement is spectacu-lar. For reference purposes the peaks that describe theatomic positions in crystalline Pd ( x -Pd) are also pre-sented. The simulated PDF for the pure is the averagevalue displayed in Fig. 1(b). A more detailed study of a -PdSi alloys is in the making (I.R. et al . Manuscript inpreparation).But what about the magnetic properties discovered inour simulations? Is this topological structure indicativeof some exciting, non-expected, electronic or magnetic FIG. 3. Comparison between total and partial experimental and simulated (solid grey) PDFs. (a) PDF for the simulated a -Pd and for the total experimental a -PdSi alloys. (b) PDF for the simulated a -Pd, partial Pd-Pd obtained by Masumoto etal . [9] and PDF for the crystalline structure. The agreement between our simulations and the experiment by Masumoto andcoworkers (as reported in ref [10]) is impressive. properties of a -Pd? If we calculate the number of nearestneighbours (nn) by integrating the area under the firstpeak of the PDF, we could infer that something is goingon since it is smaller than 12, the number of nn in thecrystalline fcc phase. This, together with the overflowof electrons from the d shell to the sp shells may be anindicator of an unexpected behaviour. However, sinceidentifying unambiguously the cutoff value to calculatethe nn in amorphous metals is a controversial subject[10] we opted for a complementary, direct approach in amanner similar to our previous calculations on bismuth[16, 17], and obtained the densities of electronic stateswith α spins and with β spins to see if they indicatea net magnetic moment, and they do, Fig. 4(a). Forthis we also used CASTEP [12] in the suite of codes ofMaterials Studio [11]. III. RESULTS AND DISCUSSION
To corroborate our results, we did some testing asfollows. We calculated the average energy per atom,magnetic and non-magnetic, and we found that thenon-magnetic value was -797.418 eV/atom and for themagnetic structure was -797.438 eV/atom. The mag-netic structure is more stable and the difference of 0.02eV/atom is of the order of reported results for some sili-con phases, 0.016 eV/atom in going from silicon diamond,the stable phase, to hexagonal diamond [18]. We also car-ried out some ab initio computational calculations for thecrystalline unit cells of nickel (fcc) and iron (bcc), bothwith zero spin and with non-zero spin initially, using thesame code (CASTEP) and the same parameters as for thepalladium jobs, to test our results and procedures. Whenthe initial spin was zero, our code and our approach led to non-magnetic results for both materials; this suggestedthat a magnetic trigger may be needed. When values ofspin of 1 and 2 µ B per atom were assigned to nickel and 1and 4 µ B to iron, the energy optimization run gave a netmagnetic moment of 0.68 µ B per atom of nickel, for bothruns, and 2.47 µ B per atom of iron, for both runs. Com-pare these results to experiment: 0.61 µ B per atom forNi [19] and 2.22 µ B per atom for Fe [20]. We performedsimilar runs for a unit cell of gold and found no mag-netism with or without an initial magnetic trigger. Wealso ran amorphous 216-atom supercells of copper, silverand gold, with and without an initial magnetic trigger,and found no remnant magnetism. This indicates thatif our procedure is applied to all these materials it leadsto the expected behaviour. We conclude that all theseresults validate our findings for amorphous palladium.Our electronic calculations indicate that the overflowof electrons, invoked for crystalline Pd, exists also for theamorphous 5s, 5p and 4d states: 4d − x (5s5p) x ; how-ever, we claim that the spin band splitting in the absenceof a magnetic field will be more preponderant in amor-phous Pd than in crystalline Pd and that the energy bal-ance ∆ E = K (1 − U N ( E F )) [with K = (1 / N ( E F ) δE and U = µ µ B λ (see Ref. [1] p. 145)] will now be ∆ E = K (1 − U N ( E F ) − V f B ) where V indicates the contribu-tion to the magnetic splitting of the unsaturated bonds f B in the amorphous. This heuristic argument wouldlead us to a modified Stoner criterion for the stability ofthe a -Pd magnetic phase: [ U N ( E F ) + V f B ] ≥
1, andthe spontaneous ferromagnetism is possible for smallervalues of
U N ( E F ).To quantify the traditional Stoner criterion, U N ( E F ) ≥
1, for the spontaneous spin band split-ting we need to obtain the product
U N ( E F ) from ourcomputational results. Here we must mention that in FIG. 4. Calculated densities of states for our three ab initio simulated supercells of a -Pd. (a) for α and β spins; a non-zeromagnetism appears when the two types of spins are contrasted, indicating a net magnetic moment. (b) for the non-magneticstate (the number of unpaired electrons is set equal to zero at the start of a single point energy calculation). The average issolid grey. The insets show the details at the Fermi level.TABLE I. Energy, magnetism and Stoner criterion( UN ( E F )) for the three amorphous palladium supercells studied.System Total energy per atom (cid:0) eVatom (cid:1) AMM ( µ B ) ∆ E (cid:0) eVatom (cid:1) δE (cid:0) eVatom (cid:1) UN ( E F )Magnetic Non-magnetic a -Pd -I -797.4391 -797.4178 0.45 -0.021 0.22 1.50 a -Pd -II -797.4323 -797.4178 0.45 -0.015 0.23 1.31 a -Pd -III -797.4427 -797.4189 0.44 -0.024 0.23 1.50Average -797.4380 -797.4181 0.45 -0.020 0.23 1.44 what follows, our numerical values are a consequenceof the parameters and approximations used in oursimulations; in particular, the use of the PBEsol func-tional, and as such the values reported herein may differfrom others where different functionals are used. Thevariations of energetic and geometrical results dependon the functionals used in atoms and molecules [21],although no magnetism is considered in this reference.This comment also applies to the results reported inTable 1.First, we start with the equation for the total energychange between the magnetic and non-magnetic states∆ E , fifth column in Table 1 (see ref [1] p. 146):∆ E = 12 N ( E F )( δE ) [1 − U N ( E F )] , (1)where N ( E F ) is the non-magnetic result (obtained bysetting the number of unpaired electrons equal to zeroat the outset of an energy calculation) and δE is thedifference between the highest energies for the magneticand non-magnetic free electron gas. The average energydifference is ∆ E = 0.02 eV atom − as shown in Table I. The product U N ( E F ) then becomes: U N ( E F ) = 1 − (cid:20) EN ( E F )( δE ) (cid:21) . (2)To calculate δE we first obtain the value of the pro-portionality constant γ in the expression for the densityof states for the free electron gas in three dimensions N ( E ) = γ √ E by requiring that the integral from thebottom of the band to E F ( E F = 7.87 eV for the non-magnetic state in Figure 4(b)) integrates to 10 stateseV − atom − . (cid:90) E F N ( E F ) dE = (cid:90) E F γ √ EdE = 10; (3)therefore the proportionality constant becomes γ = 0.68eV − / . Next we evaluate the areas under the spin-upand spin-down curves in Figure 4(a), 4.78 states per atomfor beta and 5.22 states per atom for alpha, and thenmap them onto the free electron parabola to obtain theunbalance at the Fermi energy, δE .Once we have these results and the total density ofstates at the Fermi level for the non-magnetic state (1.78states eV − spin − atom − in Figure 4(b)) we then ob-tain an average value of U N ( E F ) = 1.44 which, beinglarger than 1, satisfies the Stoner criterion (Table I). IV. CONCLUSIONS
The results reported indicate that the magnetic state ismore stable than the non-magnetic. Also, we generatedan amorphous structure for palladium that agrees withavailable experimental, partial, results. By looking at thedensities of electronic states we conclude that amorphousPd continues being a metal; in fact, a metallic glass. TheStoner criterion holds and therefore we surmise that theamorphous phase is an itinerant ferromagnet. So thatthe validity of our results can be assessed, the 1.44 valueobtained for
U N ( E F ) should be compared to those foundfor iron: 1.43 [22], or nickel: 2.03 [22], or even crystallinepalladium: 0.78 [22]. These findings may open a novelfield in the magnetism of defective metals such that, whenmacro defects are considered like pores or voids, it maybe useful in industry to produce light weight strong mag-nets. Evidently, no calculation can force a material to behave in a certain manner, so the final judge is the ex-periment. Recent experimental advances, commented inRef [23], discuss the possibility of obtaining pure amor-phous metals and these efforts may well be the begin-ning of a whole new field. Other comments that shouldbe kept in mind when dealing with simulations are thoseof Ref [24] to avoid some of the pitfalls discussed there.We believe we have been extremely careful not to forcethe simulations to produce a specific outcome. Finally,since our results are obtained for zero temperature, thismagnetism may exists at low temperatures as long as Pdremains amorphous. ACKNOWLEDGMENTS
I.R. and D.H.R. acknowledge CONACyT for support-ing their graduate studies. A.A.V., R.M.V. and A.V.thank DGAPA-UNAM for continued financial support tocarry out research projects under grant IN104617. M.T.V´azquez and O. Jim´enez provided the information re-quested. Simulations were partially carried out in theComputing Center of DGTIC-UNAM. [1] S. J. Blundell,
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