Magnetic nano-grains from a non-magnetic material: a possible explanation

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng OPEN ACCESS Magnetc nano-grans from a non-magnetc materal: a possble explanaton To cte ths artcle: Endre Kovács et al 013 IOP Conf. Ser.: Mater. Sc. Eng Vew the artcle onlne for updates and enhancements. Related content - Exact results for non-ntegrable systems Zsolt Gulács - Quadratc operators used n deducng exact ground states for correlated systems:ferromagnetsm at half-fllng provded by a dspersve band István Chalupa and Zsolt Gulács - Flat band ferromagnetsm wthout connectvty condtons n the flat band Mklos Gulács, György Kovács and Zsolt Gulács Ths content was downloaded from IP address on 8/04/018 at 00:4

2 Magnetc nano-grans from a non-magnetc materal: a possble explanaton Endre Kovács 1, Réka Trencsény, and Zsolt Gulács 1 Department of Physcs, Unversty of Mskolc, Mskolc, Hungary Department of Theoretcal Physcs, Unversty of Debrecen, Debrecen, Hungary E-mal: fzendre@un-mskolc.hu Abstract. Based on postve semdefnte operator propertes, an exact ground state soluton s deduced for a D Hubbard model wth perodc boundary condtons on small samples. The obtaned ferromagnetc behavor s used as a possble explanaton of the ferromagnetsm occurrng n nano-samples made of non-magnetc but metallc materals. 1. Introducton It s known that f a sample s constructed from non-magnetc metal (for example Au), and the sze of the obect s decreased to nanoscale values, the materal can become ferromagnetc [1,]. Below we provde a possble explanaton for ths effect, whch requres only Coulomb repulson between tnerant carrers, closed surface and quantum mechancal effects. For ths reason we analyze a two dmensonal L L square lattce at arbtrary but fnte L sze wth perodc boundary condtons n both drectons, contanng tnerant electrons. The Coulomb repulson n ths many-body system s screened by the tnerant system, consequently s of short-range type, and hence s taken nto account for smplcty by the on-ste Coulomb repulson alone. Below ths Hubbard system s solved exactly for the ground state n a restrcted regon of the parameter space, obtanng a ferromagnetc soluton for small L values. The soluton procedure (see [3-7] for detals) s based on postve semdefnte operator propertes. The technque frst transforms n exact terms the Hamltonan Ĥ of the system n a postve semdefnte form H ˆ = P ˆ + C where ˆP s a postve semdefnte operator whle C s a constant scalar. Snce the spectrum of ˆP s bounded from below by the zero mnmum egenvalue, n the second step one deduces the exact ground state Ψ g by constructng the most general wave vector whch satsfes the relaton Pˆ Ψ g = 0. The ground state energy becomes Eg = C. The ground state turns out to be ferromagnetc f the number of electrons N ( L, L) and has the form Ψ ˆ g = B, 0 where 0 represents the bare vacuum state, and ˆ, B are operators whch extend along the whole system holdng a specfc vortex structure, beng bult up from lnear combnatons of creaton Ferm operators wth fxed spn proecton actng on dfferent stes of the lattce. The parameter space regon where the soluton emerges s not severely restrcted, and flat bands n the bare band structure are not present. The perodc boundary condtons n two dmensons and both drectons lead to a closed surface whch reproduces a metallc gran, whch, because of the Coulomb repulson between Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, ournal ctaton and DOI. Publshed under lcence by Ltd 1

3 the carrers, holds moble electrons only on ts surface. Ĥ has only on-ste nteracton terms and fxed hoppng matrx elements, hence s not senstve to foldng and not requres a 6 specfc shape of the closed surface. If the gran s macroscopc ( L ~10 ), the concentraton 6 range ( ρ = L/ L ~1/ L) where the effect emerges ρ = 10 ~ 0 s practcally mssng. But for nanoscale, for example L=10, leadng to ρ = 1/10, the effect s observable, and the nano-gran behaves as a ferromagnetc obect. The remanng part of the paper presents the detals of the calculaton as follows: Sect.. descrbes the used Hamltonan, Sect. 3. shows how the transformaton of Ĥ n postve semdefnte form s made, Sect. 4. presents the deduced ground state, and fnally Sect. 5. contanng the summary closes the presentaton.. The Hamltonan We consder the Hubbard model on a D Bravas lattce wth prmtve vectors x, y and N = L L Λ lattce stes gven by H ˆ = H ˆ 0 + H ˆU, where Hˆ = ( t cˆ cˆ + t cˆ cˆ + t cˆ cˆ + t cˆ cˆ + H. c.) Here t, t x ( y) drecton, and 0 x + x,, y + y,, y+ x + x+ y,, y x + y x,,, H ˆ ˆ ˆ, > 0 U = U n n U.,, are the nearest neghbor hoppng ampltudes along the bonds n the x, (y) t (, t y+ x y x ) are the next nearest neghbor hoppngs taken along cell dagonals n y+ x,( y x) drectons, whle Hˆ U represents the Hubbard nteracton. part Λ k Usng the cˆ, = (1 / NΛ ) e ĉk, Ĥ 0 N = 1 + kx * kx + ky * ky + k ( y+ x) * k ( y+ x ) x x y y y+ x y+ x ) + k( y x) * k( y x) + ( t )] ˆ ˆ y xe + ty xe ck, ck,. () For the smplest symmetrc case t 1 t x = t y, t t y+ x= ty x, we obtan the followng dsperson relaton: ε = t (cos k + cos k ) + 4t cos k cos ky, (3) (1) expressons for the Fourer transforms, the knetc has the k-space form Hˆ = [( t e + t e ) + ( t e + t e ) + ( t e + t e 0 k, whch, except for the trval case k 1 x y x t = t = 0, cannot be made flat The postve semdefnte transformaton of the Hamltonan Let us consder on each ste of the lattce the block operator (or cell operator) defned on the cell constructed on the ste contanng the stes (, + x, + x+ y, + y ): ˆ ˆ ˆ ˆ ˆ +, (4) A, = ac 1, + a c + x, + a3 c + x + y, + a4c y, where the a n coeffcents are defned at the ste denoted by n nsde the cell (see Fg.1).

4 +y n=4 n=3 t y t y x t y+x n=1 t x +x n= Fg. 1: The notatons used n defnng the Â, operator: The presented cell s defned at the ste, the x, y are the prmtve vectors of the lattce, n denotes the stes nsde the cell, tx, ty are the hoppng matrx elements connectng nearest-neghbor stes, whle the next nearestneghbor hoppngs t and t are present along dagonals plotted by the dashed lnes. y+ x y x We note that snce U>0, the Hubbard term s automatcally postve semdefnte. Now the Hamltonan from (1) becomes ˆ Hˆ = Pˆ KNˆ = A Aˆ + Hˆ KNˆ (5),,, f the followng matchng condtons are satsfed: * * * tx = aa1+ a3a4, ty+ x = a3a1, (6) * * * ty = a4a1+ a3a, ty x = a4a. Besdes, the K coeffcent n (6) s gven by K = a 1 + a + a3 + a4. (7) We have found several dfferent solutons of ths system of complex nonlnear equatons. Here we present only the smplest one, the symmetrc sotropc case. t t = t = a, U 1 x y 1 y+x y-x 1 t t =t = a, ( ) where the condtons { x yy xy x} t > 0,, +, and t = t 1/ must hold. One can see that these condtons are not at all unrealstc and not too restrctve. The ground state energy E = 4tN. s g 1 4. The ground state Now we analyze the Hamltonan (5), and look for a ground state of the form N Ψ = ˆ 0 g D,. (8) 3

5 If such a state exsts, n order to provde frst term of Ĥ ), the Â, operator must satsfy A, A,, ˆ ˆ Ψ g =0 (e.g. to be n the kernel of the { Aˆ, ˆ } (9), D, = 0 for all values of all ndces. Furthermore, the wave functon (8) s n the kernel of Hˆ U f there are no doubly occuped stes n the system. Now we try to deduce the possble expresson of the ˆD operator. Startng the deducton of the ˆD operator, one must pay attenton to the followng aspect: Beng only one type of fermonc operator present n Ĥ, namely ĉ,, n order to obtan a ˆD satsfyng (9), all operators Â, must share wth ˆD zero, or at least two lattce stes. The man nformaton here s that for all ndces, the Â, operator must not touch ˆD n a sngle pont. Ths s possble only f ˆD s extended along the whole system. To satsfy ths condton, let us consder a general expresson for the ˆD operator collectng addtve contrbutons from all lattce stes of the system. The mathematcal expresson of the ˆD operator then becomes L L, = x, c ( 1) + ( 1), = 1 = 1 Dˆ ˆ x y, (10) where xy, are the prmtve vectors, and n the x, coeffcents the s the row ndex, whle s the column ndex. Snce n (9) one can have L dfferent Â, operators for a fxed spn proecton, n order to deduce the x, coeffcents from (9), one obtans L dfferent equatons. The frst one for example s the followng: a1x 1,1 + a x 1, + a3x n, + a 4x n,1 = 0 Usually the solutons can be deduced for fnte L, and the obtaned result must be extended to arbtrary large L. We show n a smple symmetrc and sotropc case that ths can be performed, ndeed. One ˆD operator obtaned for ths case at L=1 s presented n Fg.. As one can see from ths fgure, the dsplacement of partcles nsde ˆD follows vortex lnes. These lnes are dsplaced around 4 equvalent vortex centers (denoted by dotted squares n the fgure) whose poston can serve n denotng dfferent ˆD operators. For example the bottom-left ste of a vortex center can serve for the ndex of the operator ˆD. The spn ndex of the operator ˆD s the same on all stes. Modfyng the poston of the vortex center, one fnds new ˆD operators. 4

6 Fg.. A ˆD, operator deduced at L=1. Black dots at an arbtrary ste are representng whte dots at a ste are representng must be added provdng the deduced ĉ, ˆD ĉ,, and + operators. All these contrbutons presented n the fgure squares are representng 4 equvalent plaquettes n the constructon of. The dotted lnes represent a gude for the eyes. The dotted extenson of the soluton to an arbtrary large system (for even L) s possble. In obtanng the mathematcal expresson of the ˆD, ˆD. It can be seen that the operator from Fg.., one takes nto consderaton that black dots placed at an arbtrary ste are representng ĉ, operators, whle whte dots at an arbtrary ste are representng + ĉ, operators. Addng all contrbutons from Fg. one obtans the mathematcal expresson of the plotted ˆD operator. Fg. shows that the ˆD soluton obtaned for a 1 1 lattce can be generalzed to an arbtrary large system (for even L). Dfferent lnearly ndependent ˆD operators can be obtaned by modfyng the poston of the vortex center. Snce for all (even) L, four equvalent vortex center postons exst, at frst 5

7 vew t seems that N Λ / 4 lnearly ndependent ˆD operators exst. However, f we drectly check the lnear ndependence, t turns out that only L of them are lnearly ndependent. The product of all lnearly ndependent ˆD operators leads to touchng ponts between all components of the product. To enter also n the kernel of ˆ H U, the spn ndex for all ˆD operators wll be fxed to the same value =, hence the ground state becomes ˆ, (11) Ψ g = D 0, whch represents a saturated ferromagnetc state. 5. Summary By applyng a technque based on the propertes of postve semdefnte operators we analyzed a two dmensonal system wth perodc boundary condtons, where the Hubbard on-ste Coulomb repulson acts between the tnerant electrons. For small samples ferromagnetsm has been found whch dsappears when the sze of the system ncreases. The results provde a possble explanaton for the emergence of ferromagnetsm on metallc nanograns bult up from a non-magnetc materal. The deduced result does not requre rgorous sphercal surface, nor spn-orbt nteracton: t demands only tnerant electrons on a closed surface, Coulomb repulson between the carrers, and quantum mechanc many-body behavor. ACKNOWLEDGEMENTS We kndly acknowledge fnancal support of the followng contracts: for E.K. TAMOP B-10//KONV , for R.T. TAMOP-4../B-10/ and for Z.G. OTKA-K-10088, TAMOP-4..1/B-09/1/KONV , and Alexander von Humboldt Foundaton. References [1] P. Crespo, R. Ltran, T. C. Roas, et al. Phys. Rev. Lett. 93, (004) [] S. Baneree, S. O. Raa, M. Sardar et al. Cond-mat arxv: [3] Z. Gulacs, A. Kampf, D. Vollhardt, Phys. Rev. Lett. 105, (010); [4] Z. Gulacs, D. Vollhardt, Phys. Rev. Lett. 91, (003) [5] Z. Gulács, A. Kampf, D. Vollhardt, Phys. Rev. Lett. 99, (007). [6] R. Trencseny, K. Gulacs, E. Kovacs, Z. Gulacs, Ann. der Phys (Berln) 53, 741 (011). [7] R. Trencsény, E. Kovács, Z. Gulács, Phl. Mag. B, 89, 1953 (009). 6