Universal distribution of magnetic anisotropy of impurities in ordered and disordered nano-grains

1.5 MB·English

by  A. Szilva

Universal distribution of magnetic anisotropy of impurities in ordered and disordered
nano-grains
A. Szilva,1,2 P. Balla,2,3 O. Eriksson,1 G. Zar´and,4 and L. Szunyogh2,5
1Department of Physics and Astronomy, Division of Materials Theory,
Uppsala University, Box 516, SE-75120, Uppsala, Sweden
2Department of Theoretical Physics, Budapest University of Technology
and Economics, Budafoki u´t 8. H-1111 Budapest, Hungary
3Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,
Hungarian Academy of Sciences, H-1525 Budapest, P.O.B. 49, Hungary
4BME-MTA Exotic Quantum Phases ’Lendu¨let’ Group, Institute of Physics,
5
Budapest University of Technology and Economics, H-1521 Budapest, Hungary
1
5MTA-BME Condensed Matter Research Group, Budapest University of
0
2 Technology and Economics, Budafoki ut 8., H-1111 Budapest, Hungary

n Weexaminethedistributionofthemagneticanisotropy(MA)experiencedbyamagneticimpurity
a embedded in a metallic nano-grain. As an example of a generic magnetic impurity with partially
J filled d-shell, we study the case of d1 impurities imbedded into ordered and disordered Au nano-

1 grains, described in terms of a realistic band structure. Confinement of the electrons induces a
2 magnetic anisotropy that is large, and can be characterized by 5 real parameters, coupling to the
quadrupolar moments of the spin. In ordered (spherical) nano-grains, these parameters exhibit

] symmetrical structures and reflect the symmetry of the underlying lattice, while for disordered
l
l grains they are randomly distributed and, – for stronger disorder, – their distribution is found to
a
be characterized by random matrix theory. As a result, the probability of having small magnetic
h
anisotropies K is suppressed below a characteristic scale ∆ , which we predict to scale with the
- L E
s number of atoms N as ∆E 1/N3/2. This gives rise to anomalies in the specific heat and the
e susceptibility at temperature∼s T ∆ and produces distinct structures in the magnetic excitation
E
m ∼
spectrum of the clusters, that should be possible to detect experimentally.
.
t
a
m Magnetic thin films and nano-sized objects are es- ferent mechanisms. In metallic compounds of heavy el-
sential ingredients for high-density magnetic recording. ements with strong SO interaction, the geometry of the
-
d Magnetic nanoparticles, in particular, are considered as sample is imprinted into the spin texture of the conduc-
n
most likely building blocks for future permanent mag- tion electrons’ wave function. This spin texture varies
o
nets [1–3]. Similar to molecular electronics devices [4] or in space close to the surface of the sample, and induces
c
[ thin metallic layers [5, 6], spin-orbit (SO) coupling plays a position dependent magnetic anisotropy for the mag-

an essential role in nano-particles: by restricting the free netic dopants. The corresponding host-induced mag-
1
motion of the magnetic spins and eventually freezing netic anisotropy (HSO mechanism), intensively studied
v
6 them [7], it enables spins to store magnetic information. in atomic-scale engineering, is presumably at work in
4 Understanding the behavior of the magnetic anisotropy magnetically doped noble metal samples, where it gives
3 in systems with quantum confinement is hence of crucial rise to a relatively short-ranged confinement-induced
5
importance for nanoscale magnetic materials science. magnetic anisotropy close to sample surfaces [7]. Much
0
. SO coupling induced magnetic anisotropy (MA) ap- stronger and longer-ranged anisotropy can, however, be
1 pears to be surprisingly large in certain nanoscale and generated by the local SO coupling at the magnetic
0
mesoscopic structures. A sterling example, where con- dopant’s d or f-level [10, 11] in case of magnetic impuri-
5
1 finement induced MA provides explanation for the ob- ties with a partially filled d or f shell, respectively [10].
: servation, is the suppression of the Kondo effect in thin Since, in this case, the spin of the magnetic ion is en-
v
i films and wires of certain dilute magnetic alloys [5]. As tangled with the orbital structure of localized f and d
X revealedbyaseriesofexperimentsonmagneticallydoped states, and couples very strongly to Friedel oscillations,
r thin metallic films [5, 6], SO coupling combined with a leading to the emergence of a strong MA (LSO mech-
a
geometricalconfinementoftheelectrons’motioninduces anism). While the HSO mechanism appears to be too
a ’dead layer’ in the vicinity of the surface, where the weak to explain the thin film experiments, the stronger
motionoftheotherwisefreespinsisblockedbyMA.The andmoreslowlydecayinganisotropyinducedbytheLSO
thickness d of this ’dead layer’, consistently explained in mechanismseemstogiveaconsistentexplanationforthe
terms of surface induced spin anisotropy [5, 6], depends experimentalobservations[10–12],andappearstobethe
on the particular host material and dopants used, but it dominant mechanism for SO coupling induced MA in
can be unexpectedly large, in the range of d 100˚A. confined structures.
∼
Inconfinedgeometries,spin-orbit(SO)interactioncan The surface-induced MA has been thoroughly studied
induce magnetic anisotropies by two fundamentally dif- in thin films and in the vicinity of surfaces. Surpris-
2
ingly little is known, however, about the structure and Following Anderson, we consider hybridization of the
size of confinement induced MA in nano-grains. Here deep D3/2 multiplet only with s-type host electrons,
we therefore investigate the LSO mechanism in metal- since these latter dominate the density of states near
lic grains and demonstrate that symmetrically ’ordered’ the Fermi-energy (see Ref. [16]). Local cubic sym-
nano-grains and nano-grains with random surfaces show metry implies, however, that only linear combinations
verydifferentbehaviors. Inorderednano-grains,theMA of neighboring s orbitals, transforming as j 3/2
∼
constants exhibit regular structures reflecting the sym- can hybridize with the deep D3/2 states. The proper
metry of the grain. Different atomic shells of the grain s , s , s , s basis set has been con-
3/2 1/2 1/2 3/2
| i | i | − i | − i
behaveverydifferentlyfromthepointofviewofmagnetic structed in Ref. [9, 12] and is reproduced in the Sup-
anisotropy,whichdisplaysFriedel-likeshell-to-shelloscil- plemental material, Ref. [16]. Considering then charge
lations. Adding atoms to an ordered grain and thereby fluctuations to the d0 state and performing a Coqblin-
making its surface disordered, however, changes this pic- Schrieffertransformation[20,21],wefinallyarriveatthe
ture completely: in such ’disordered’ grains, the conduc- following simple exchange Hamiltonian,
tion electron’s wave function becomes chaotic, and the
=J s s m m . (1)
distribution of MA parameters become gradually more HLSO †m m0| 0ih |
andmorerandom. TheMAdistributionisthenfoundto mX,m0
be fairly well captured by random matrix theory, and to Here the m refer to the states 3,1, 1, 3 of
be almost independent of the magnetic ion’s position. the impuri{ty|, ain}d s creates appropria{t2e h2os−t2ele−ct2r}ons,
†m
In the present work, we shall demonstrate these char- while J denotes the strength of the effective exchange
acteristic properties by focusing on the simplest case of coupling (see also Ref. [16]).
amagneticimpurityinad1 configurationembeddedinto TohandletheexchangeinteractionJ,wecanuseadi-
an fcc Au nano-grain host of 100-400 atoms. This model agrammatic approach similar to Ref. [9]. The dominant
system captures the generic properties of most magnetic contribution to the MA is, however, simply given by the
impurities and hosts, and allows us to study the roles of Hartee term, generating the effective spin Hamiltonian,
local and host SO interactions simultaneously. We con-
struct the nano-grains by placing Au atoms on a reg- HL = Kmm0|m0ihm|
ular fcc lattice starting from a central site, and then mX,m0
adding ’shells’, defined as groups of atoms that trans-
with the anisotropy matrix K expressed as
form into each other under the cubic group (O ). We mm0
h
refer to nano-grains with only filled shells as ordered (or K =J s s =J εF dερL (ε). (2)
spherical) nano-grains, while nano-grains with partially mm0 h †m m0i mm0
Z−∞
filledoutmostshellsshallbereferredtoasdisordered (or
Here ρL(ε) denotes the local spectral function matrix of
non-spherical) nanoparticles. We also define the core of
thesymmetryadaptedhostoperators,s ,andε stands
the grain as the group of atoms having a complete set †m F
fortheFermi-energy. Inpractice,weevaluatetheintegral
of first neighbors. To describe the electronic structure of
(2) in terms of the Green’s functions of the host [23].
Au nano-grains, we use a tight binding model with spd
Althoughdisorderednano-grainsdonotpossessspatial
canonical orbitals and incorporate SO coupling of the
symmetries,timereversal(TR)symmetryisstillpresent,
host atoms non-perturbatively. More technical details
andimpliesthat,apartfromanunimportantoverallshift
can be found in the Supplemental Material [16].
K , the anisotropy matrix K can be parametrized in
To investigate the local SO-induced anisotropy, we 0 mm0
terms of five real numbers, K (µ=1...5),
µ
shall use the approach of Ref. [9], and account for local
correlationsonthemagneticimpuritybymeansofagen- K K K i K K i 0
1 3 5 2 4
− −
eralized Anderson model [17], which we embed into the K +K i K 0 K K i
 3 5 − 1 2− 4  ,
Au grain described above. Similar to Anderson’s model, K +K i 0 K K +K i
2 4 1 3 5
− −
ourimpurityHamiltonian(theso-calledionicmodel[18])  0 K +K i K K i K 
 2 4 − 3− 5 1 
contains three terms: the impurity term, the conduction  (3)
electron, and the hybridization terms (see Ref. [16]). In which we shall refer to as the LSO-MA parameters. We
the ground state d1 configuration, by Hund’s third rule, note that the absence of SO coupling on the host atoms
the strong local SO interaction aligns the angular mo- further simplifies the structure of HL, and the matrix
mentum of the d-electron antiferromagnetically with its elementsK , K andK vanishincasetheupanddown
3 4 5
spin, thus forming a D3/2 spin j = 3/2 multiplet. This spin channels do not mix in the host.
multiplet remains degenerate in a perfect cubic environ- TheMAmatrixinEq. (3)hastwoKramersdegenerate
ment, and – in group theoretical terms – it transforms eigenvalues, λ = λ , whose splitting can be used to
+
according to the four-dimensional Γ double representa- define naturally the−m−agnetic anisotropy constant as
8
tion of the cubic point group [19].
Next, we need to embed this impurity into the host. KL ≡(λ+−λ−)/2 = µKµ2 . (4)
q
P
3

Shell0
Shell1
2.50 Shell2
Shell3 5.0
Shell4
Shell5
Shell6
1.25
2.5
V] V]
me 0.00 me 0.0
[2 [2
K K
-1.25 -2.5
-5.0
-2.50
-2.50 -1.25 0.00 1.25 2.50 -5.0 -2.5 0.0 2.5 5.0
K1 [meV] K1 [meV]
FIG. 1: (color online) Left: Anisotropy parameters in the E-plane for an ordered grain of 225 atoms (87 core atoms). Right:
Anisotropies in the E-plane in case of 100 disordered Au nano-grains. An ordered cluster is created by 225 atoms that fill
completelythefirstsixshellsofthefccstructure,andwith11extraatoms,randomlyplacedontothenextshell. Thetriangular
structure of the anisotropy parameter distribution can still be observed.
If the magnetic impurity is placed in an ordered (spher- follows. The left panel of Fig. 1 shows the computed
ical) nano-grain then the MA matrices can be different (K ,K ) values, plotted in the E-plane for an ordered
1 2
for different sites even if they belong to the same shell. grain of 225 atoms (87 core atoms). Throughout this
Their trace, eigenvalues and, therefore, the MA constant work, we use J = 0.25 eV, a value consistent with a
should,however,bethesameforallsiteswithinthesame Kondo temperature below 0.1 K. Different colors denote
shell due to the underlying (O ) symmetry of the grain, the MA parameters of clusters with magnetic impurities
h
as indeed confirmed by our numerical simulations, dis- placed on the different shells. Similarly, the anisotropy
cussed below (for additional information, see Ref. [16]). constantsK ,K ,andK (showninFig. 4inRef. [16]),
3 4 5
To find connections between the elements of K for inducedbytheSOinteractiononthehost Auatoms,are
mm0
anorderednano-grain,weexpressEq. (3)inamultipolar relatedforatomsonthesameshell,andshowregularpat-
basis. TimereversalsymmetryimpliesthatKmm0 canbe terns in a three dimensional space, the T2-space. These
expressed solely in terms of even powers of the j = 3/2 parameters are, however, smaller by about one order of
spin operators, Jx,Jy,Jz. In fact, the parameters in (3) magnitude compared to the parameters K1,2, implying
couple directly to the usual 5 (normalized and traceless) that the MA constant, Eq. (4), is dominated by the E-
quadrupoleoperatorsallowedbytimereversalsymmetry, type parameters.
Q ,...,Q proportionalto2J2 J2 J2,J2 J2,J J +
1 5 z− x− y x− y x z Next, let us examine the distribution of the magnetic
J J , J J +J J and J J +J J , respectively. The
z x x y y x y z z y anisotropy constants in case of disordered nanoclusters.
local Hamiltonian can be simply expressed in terms of
First we created 100 disordered nanoclusters by adding
these as
11extraatomstoan225-atomorderedcluster,andplac-
ing them randomly on the next shell of 24 possible sites.
HL = K Q , (5)
µ µ We then calculated the MA parameters on all core sites
Xµ foreverynanoparticle. InFig.1(right)weshowthe8700
E-plane parameters obtained this way. Different colors
with the coefficients K of the Q-matrices forming a 5-
µ
represent data from different shells. Small ’clouds’ are
dimensionalvector. Undercubicpointgrouptransforma-
observed with obvious remains of the three-fold symme-
tions, the first two components, (Q ,Q ) and the last
1 2
∼ try,butthereisnostrictlyorderedstructureleftanymore
three components, (Q ,Q ,Q ) transform into each
3 4 5
∼ in the E-plane.
other according to the E and T representations of the
2
cubic point group, respectively [19]. Correspondingly, We then increased the structural disorder of the nano-
for atoms on the same shell of an ordered grain, the grains further, and added 25 extra atoms to an ordered
anisotropy parameters K transform into each other, cluster of 225 atoms, by placing them randomly on the
1,2
and form regular patterns of triangular symmetry in next three shells. As shown in the inset of Fig. 2, for
the (K ,K )-plane, referred to as the E-plane in what these strongly disordered clusters the distribution be-
1 2
4
comes almost isotropic in the E-plane, and the trian-
gular symmetry is almost entirely lost. The main panel
of Fig. 2 shows the radial distribution of the magnetic -1V] 5.0
anisotropy parameters, K2+K2 in the E-plane. The e0.3
1 2 m 2.5
ottihboesnecsrovomefdpaodsniimsetnrptilsbeuoGftiKaounEssa≡igaprn(eKetsh1e,voKerry2y,)whwahevellerewinwidtehepateshsnuedmepnertetdahniacdt- of MA [0.2 K[meV]2-02..05
Gaussian distribution, n -5.0
o
p(KE)∼e−K2E/∆2E , (6) stributi0.1 (cid:1)E (cid:2) 2.37meV -5.0 -2.5K10[m.0eV]2.5 5.0
with the E-plane anisotropy scale ∆ defined as ∆2 Di
E E ≡ 0
K2 . We find that in these disordered grains the ra- 0 1 2 3 4 5 6 7
hdiaEl idistribution in the three-dimensional T -space pa- |K | [meV]
2 E
rameters can also be fitted by a similar Gaussian en-
semble, although with a smaller characteristic radius, FIG. 2: (color online) Radial distribution of the magnetic
K2 ∆ <∆ (seeRef.[16]). Theoveralldistribu- anisotropy parameters (dots) in E-plane in case of NS=50
h Ti≡ T E sampleswithN=225+25atoms. Thecontinuouslinepresents
tionoftheanisotropyK isthereforestronglysuppressed
p L the predictions of the Gaussian Orthogonal Ensemble. Inset:
at small values. In the absence of host SO coupling,
Distribution of (K ,K ) in the E-plane for these 50 nano-
1 2
it scales as p(KL) KL for small anisotropy values, grains. At this level of disorder the triangular structure is
∼
KL <∆E,whileinthepresenceofitp(KL)issuppressed almost entirely lost.
as p(K ) K 4 for K < ∆ . This implies that typ-
L L L T
∼ | |
ical sites in a disordered grain have a finite SO-induced
anisotropy of size ∆ , and of random orientation, al- theMAenergy’sdistribution. Thetypicalanisotropyval-
E
∼
most independently of their precise location within the ues,∆ 0.57THz,shiftrapidlytowardssmallervalues
E
≈
grain. (GHz) with increasing grain size.
We remark that, – even after adding a single extra The universal anisotropy distribution should be in-
atom to an ordered cluster, – the distribution of the directly observable through thermodynamic quantities,
eigenenergies of the host Hamiltonian agreed with the too. In the presence of a random distribution of
predictions of random matrix theory and, in agreement anisotropies, given by Eq. (6), we obtain a peak in the
with the experimental findings [13], exhibited level re- specific heat C(T) at T 0.78 ∆ , and a low tempera-
E
≈
pulsion according to a Gaussian symplectic (GS) level turespecificheatC(T) T2/∆2 (seeRef. [16]),turning
∼ E
spacing distribution. The observed GS distribution re- into a T5 anomaly for T ∆ . Similarly, the coeffi-
T
∼ (cid:28)
flects the chaotic nature of the electron’s wave function cient of the Curie susceptibility, Tχ, should exhibit a
∼
aswellasthepresenceofhost SOcoupling[16]. Building
upon the chaotic nature of the electron’s wave function,
oansseumcainngorbatnadinomanpleasntiemwataeveofco∆ndEu,ctuisoinngelEecqt.ro(n2)waanvde u.]3000
functions [22]. This yields the estimate, a.[2500
m
u2000
∆E ∼J (∆SO/(cid:15)F)/N3/2, pectr1500
s
withN thenumberoflatticesitesontheclusterand∆
SO n
the SO splitting of the j = 3/2 and j = 5/2 impurity- atio1000 2ΔE ≈1.14THz
levels. cit 500
The anisotropy distribution (6) has a direct impact Ex
0
on the magnetic excitation spectrum of the nano-grains.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Generating 50 (strongly) disordered nano-grains with
Frequency [THz]
225+40 atoms (87 core sites), we randomly chose and
rotated 100 nano-clusters from the 4350 different sam-
FIG. 3: (color online) Orientational and disorder averaged
ples (see Ref. [16]), and determined the magnetic im-
excitation spectrum computed for an ensemble of 1000 ran-
purities’ excitation spectrum averaging over the orienta- domly chosen and randomly oriented disordered nano-grains
tion of them. All these 100 spectra were added together, (N=225+40). Theshapeofthesignalreflectsthedistribution
andtheprocedurewasrepeatedtentimes. Theobtained of magnetic anisotropies while the small peak corresponds to
aggregated spectrum is shown in Fig. 3. The obtained transitionsbetweenthelowestKramersdoublets,splitbythe
magnetic field. Inset: spectra of 100-100 selected grains [16].
spectra are grain specific (see inset), and reflect directly
5
strong suppression below T ∆E. 209 (1991).
∼
Although the model discussed here has features that [6] J.F. DiTusa, K. Lin, M. Park, M.S. Isaacson, and J.M.
are specific, we believe that it captures many generic Parpia, Phys. Rev. Lett. 68, 1156 (1992).
propertiesofmagneticimpuritiesinametallicgrain,and [7] O.U´jsa´ghy,A.Zawadowski,andB.L.Gy¨orffy,Phys.Rev.
Lett. 76, 2378 (1996).
thusallowsonetodrawgeneralconclusions;Foranymag-
[8] L.SzunyoghandB.L.Gyo¨rffy,Phys.Rev.Lett.78,3765
netic impurity of spin J, time reversal symmetry implies
(1997).
that the leading anisotropy term is of the form (5). In
[9] L. Szunyogh, G. Zara´nd, S. Gallego, M. C. Mun˜oz, and
ordered grains, the distribution of the five parameters B. L. Gyo¨rffy, Phys. Rev. Lett. 96, 067204 (2006).
K must always reflect the underlying lattice symmetry, [10] A.Szilva,S.Gallego,andM.C.Munoz,B.L.Gyorffy,G.
µ
and for cubic lattices, in particular, the couplings K Zarand, L. Szunyogh, Phys. Rev. B 78, 195418 (2008).
E
and K are organized into triangular and cubic struc- [11] A. Szilva, S. Gallego, and M. C. Munoz, B.L. Gyorffy,
T
G. Zarand, L. Szunyogh, IEEE Trans. Magn. 44, 2772
tures,respectively. Theseparametersareexpectedtobe-
(2008).
comerandom, andtoexhibitmultidimensionalGaussian
[12] A.Szilva,Theoreticalstudyofmagneticimpuritiesinlow
distributions in sufficiently disordered grains. The sup-
dimensional systems, PhD Thesis, BME (2011).
pression of the probability of having a small anisotropy,
[13] F. Kuemmeth, K. I. Bolotin, S. Shi and D. C. Ralph,
p(KL 0)=0,aswellasthepredictedspecificheatand Nano Lett., 8 4506-4512 (2008).
→
susceptibility anomalies are also generic features, since [14] P.W.Brouwer,X.Waintal,andB.I.Halperin: PhysicalRe-
they follow simply from the presence of randomly dis- view Letters, 85, 369 (2000).
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µ
(1999).
conclusionsregardingtheSchottkyanomalyarethusgen-
[16] See Supplemental Material for the description of Au
eral, though details of the low temperature scaling of
nanoparticle host, the derivation of the LSO effective
C(T) may be system specific.
Hamiltonian,thecalculationofthemagneticanisotropy,
We owe thanks to Agnes Antal, Imre Varga and Jan theanalysisofthehostlevelspacingdistributioninterms
Rusz for the fruitful discussions. This work has been of random matrix theory, the illustration of magnetic
financed by the Swedish Research Council, the KAW excitation spectra of impurities in nano-grains, and the
foundation, and the ERC (project 247062 - ASD) and Schottky anomalous heat capacity.
[17] P.W. Anderson, Phys. Rev. 124, 41 (1961).
the Hungarian OTKA projects K105148 and K84078.
[18] L.L Hirst Adv. Phys. 27,231 (1978).
We also acknowledge support from eSSENCE and the
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SwedishNationalAllocationsCommittee(SNIC/SNAC).
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[20] A.C. Hewson: The Kondo Problem to Heavy Fermions,
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[21] B. Coqblin and J. R. Schrieffer, Phys. Rev. 185, 847
(1969).
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SO
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Universal distribution of magnetic anisotropy of impurities in ordered and disordered
nano-grains
A. Szilva,1,2 P. Balla,2,3 O. Eriksson,1 G. Zar´and,4 and L. Szunyogh2,5
1Department of Physics and Astronomy, Division of Materials Theory,
Uppsala University, Box 516, SE-75120, Uppsala, Sweden
2Department of Theoretical Physics, Budapest University of Technology
and Economics, Budafoki u´t 8. H-1111 Budapest, Hungary
3Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,
Hungarian Academy of Sciences, H-1525 Budapest, P.O.B. 49, Hungary
4BME-MTA Exotic Quantum Phases ’Lendu¨let’ Group, Institute of Physics,
5
Budapest University of Technology and Economics, H-1521 Budapest, Hungary
1
5MTA-BME Condensed Matter Research Group, Budapest University of
0
2 Technology and Economics, Budafoki ut 8., H-1111 Budapest, Hungary

n
a
J APPENDIX A - DESCRIPTION OF THE GOLD
TABLE I: The shell structure of fcc clusters. Label, a and b
1 NANOGRAIN-HOST
denote the shells where the atoms are at the same distance
2
from the center atom but cannot be transformed into each

We have defined the structure of the Au nano-grains other(underO ). N denotesthenumberofsitesinagiven
] h sh
l host of N=100-400 atoms as follows: one can speak of shell, N is the total number of atoms and Nc is the number
l
a an ordered grain when it has only filled shells around of core sites in the cluster.
h
a center atom, while nano-particles with partially filled
-
s (outmost)shellshavebeenreferredtoasdisordered nano-
me grains. Theshell isagroupofatoms(Nsh)onanfcc lat- Shell Nsh N Nc Shell Nsh N Nc
ticethattransformintoeachotherunderthecubicgroup
t. (Oh). The shell-structure of a few fcc nano-clusters is Center 1 1 0 16NNa 24 405 177
a
shown in Table I: an ordered grain built by, say, N=225 1NN 12 13 1 16NNb 24 429 201
m
atoms has 12 filled shells: a 1 center atom, 12 first-, 6 2NN 6 19 1 17NNa 24 453 225
-
d second-... and24twelfth-neighbors. Thesite(0,1.5,1.5) 3NN 24 43 1 17NNb 6 459 225
n belongs to shell 9NNa, while the site (0.5,0.5,2) is on 4NN 12 55 13 18NN 48 507 249
o shell 9NNb, though they are at the same distance from
5NN 24 79 19 19NN 24 531 249
c the origin (center atom). If a disordered grain host of
[ 6NN 8 87 19 20NN 24 555 273
N = 236 atoms has 11 atoms in the outmost (12NN)
7NN 48 135 43 21NN 48 603 321
1 shell(insteadofN =24atoms),thenonecandealwith
v C24 configurationss.hInpractice,wechooserandomlyonly 8NN 6 141 43 22NN 24 627 321
6 11 9NNa 12 153 55 23NN 48 675 369
N grains from the big configuration space (N is typ-
4 S S 9NNb 24 177 55 24NN 8 683 369
ically set as 50-100). It should be noted that one can
3
5 put extra atoms not only into the first outmost shell but 10NN 24 201 79 25NNa 48 731 393
0 we never generate ”holes” in a nano-cluster. In a given 11NN 24 225 87 25NNb 12 743 405
1. nano-grain the atoms those that have all the first neigh- 12NN 24 249 87 26NN 24 767 411
0 bors are called as core atoms (denoted by N in Table 13NNa 48 297 135 27NN 24 791 435
c
5 I). The core region is away from the surface of the nano- 13NNb 24 321 141 28NN 24 815 459
1
grain. If a nano-particle is built by N=225 atoms then 14NN 48 369 165 29NN 24 839 459
:
v ithasNc=87coraatoms(beinginthecentersiteandon 15NN 12 381 177
i six core shells).
X
The electronic structure of the Au nano-grains will be
r
a described by a tight binding (TB) Hamiltonian. The
model uses spd canonical orbitals, and the spin-orbit
(SO) coupling of the host atoms is considered non- so-called canonical basis (real spherical harmonics),
perturbatively. Specifically, the TB model uses (nearly)
α=s ‘=0
orthonormal basis functions which are localized at sites,
Rn, α=px,py,pz, ‘=1 , (2)
α=d ,d ,d ,d ,d ‘=2
xy xz yz x2 y2 3z2 1
r n;ασ = r R ασ =ψ (r R )φ , (1) − −
n α n σ
h | i h − | i −
ψ depends only on the azimuthal quantum number ‘
α
where n refers to the given site, the index α denotes the and the spin quantum number is labeled by σ = 1.
±2
2
The Hamiltonian of the noble metal host is written as
Hˆ = Hn,n0 =(ε δ δ +ξHLS )δ +tn,n0δ ,
ασ,α0σ0 α αα0 σσ0 ασ,α0σ0 nn0 α,α0 σσ0 s
n o (3) 2.5 p
d
where the dimension of the matrix is M =18×N, εα is eV) e F=7.4 eV
the so-called on-site energy parameter, 1/ 2
(
s
HαLσS,α0σ0 =hασ|L~ S~|α0σ0i, (4) state1.5
f
o
ξ is the SO coupling parameter and tn,n0are the hy- y
bridization matrix elements (or hoppingα,αin0tegrals) be- nsit 1
e
tween the different orbitals. D
0.5
We note that on-site energies ε , ε , ε and ε
s p d−Eg d−T2g
were used in case of all calculations. The hopping inte-
0
grals to first- and second nearest neighbors were consid- -2 0 2 4 6 8 10 12 14 16 18 20 22 24
Energy (eV)
ered. They depend only on the relative positions of the
sites, i.e.,
FIG. 1: (color online) The spd density of states (DOS) com-
tn,n0 =t (R R ) . (5) ponentsofanorderednano-grain(N=225)normalizedtoone
α,α0 α,α0 n0 − n atom. The calculated Fermi-energy is ε =7.4 eV.
F
The numerical values for both ε and t can be found
α α,α0
in Ref. [1]. The matrixelements of the SO coupling can
APPENDIX B - DERIVATION OF THE LSO
easily be calculated with the help of following identity,
MODEL HAMILTONIAN
1
L~ S~ = (L S +L S )+L S . (6)
+ + z z
2 − − Next we derive the local spin-orbit (LSO) model for a
d1-type magnetic impurity. In the (non-degenerate) An-
The spin-orbit coupling parameter ξ was determined
dersonimpuritymodel[3]asingleenergylevel,ε ,iscon-
from the difference of the SO-split d-resonance energies d
sideredattheimpurity. Thislevelcanbeatmostdoubly
∆E =E E , (7) occupied. Because of the Coulomb repulsion, the energy
d j=5/2 j=3/2
−
of the doubly occupied state is 2ε +U. Moreover, since
d
as derived from self-consistent relativistic (SKKR) first-
the wavefunction of the d level is not orthogonal to the
principlescalculations[2]. Thissplittingisrelatedtothe
states of the conduction band, they may be hybridized.
strength of SO coupling as
Weconsiderafreeionthathasthefollowingfourpossible
5 states: the d level is empty in state d0 , it is occupied
∆Ed ξ . (8) by an electron with spin σ in state d1| ,iand it is doubly
’ 2 | σi
occupied by electrons with opposite spins in state d2 .
For Au bulk we obtained ξ =0.64 eV. The energies of these states are ε(d0) = 0, ε(d1) =| εi
σ d
The Green’s function or resolvent operator of a nano- and ε(d2) = 2ε +U. The ground state of the ion has a
d
particle is defined as
magnetic moment if the magnetic doublet is the lowest
in energy, that is if ε(d2)>ε(d1) and ε(d0)>ε(d1), i.e.
Gˆ(z)=Gn,n0 (z)=(z Hˆ) 1 (9) σ σ
ασ,α0σ0 − − if −21U < εd + 12U < 12U. When U is large, no double
occupancy happens, and the spin fluctuates.
that can be written as
The so-called ionic model is a possible generalization
M v v of the Anderson model [4]. The Hamiltonian contains
Gˆ(z)= | iih i| , (10)
z ε threetermsasusual: theimpurity,thehost (conduction)
i
Xi=1 − electron, and the hybridization terms. If the multiplet of
where{εi}and{|vii}standfortheeigenvaluesandeigen- theimpurityionisdenotedby|n,mni,wherenindicates
vectros of the Hamiltonian Eq. (3), respectively. the number of electrons in the shell and mn the set
{ }
Finallywedefinethedensityofstates(DOS)asfollows, of quantum numbers characterizing the multiplet, and
the corresponding energy is denoted by E , then the
1 n,mn
n(ε)= limTr Gˆ(ε+iδ) Gˆ(ε iδ) . (11) impurity term can be written as
−2πiδ 0 − −
→ (cid:16) (cid:17) = E n,m n,m . (12)
The numerically calculated values are shown in Fig. 1 Himp n,mn| nih n|
for an ordered nano-grain. The calculated Fermi-energy nX,mn
is ε = 7.4 eV, at the Fermi-energy the s contribution If the impurity has only a single non-degenerate d level
F
dominates the DOS. and the Coulomb interaction U is sufficiently large, so
3
that the doubly occupied configuration has high energy, find the relation between standard Γ basis
8
so that the only relevant configurations are n = 0 or
n = 1, then the Hamiltonian of the ionic model, Hionic, |s3/2i
can be approximated as s
 | 1/2i  , (16)
s
1/2
| − i
E0|0,0ih0,0|+ E1,m|1,mih1,m|+ ε~ks~†k,ms~k,m+ |s−3/2i
Xm ~Xk,m and the tensor-product basis D σ , where δ = x2
{ δ| i} −
+ V~k|1,mih0,0|s~k,m+V~k∗s~†k,m|0,0ih1,m| , y2 or 2z2 − x2 − y2 and σ =↑,↓. We should find the
~Xk,m(cid:16) (cid:17) transformation matrix between the two basis as follows,
(13)
s =Q D σ . (17)
{| mi} { δ| i}
wheremrunsover2j+1values. Incaseofj = 1 theU = To find the proper Q, we compared the action of a rota-
2
Anderson model is recovered, see Ref. [5], Sec. 1.9. tion around the z and x axis by angle π in both bases
∞ 2
In Eq. (13) the operator s† creates a host conduction and obtained
~k,m
electron with wavenumber ~k, pseudospin m and energy 0 1 0 0
−
ε . V -s denote the s-d hybridization matrix elements.
~k ~k Q=0 0 1 0  , (18)
As we mentioned in the main paper, the desired ef- 0 0 0 1
fective Hamiltonian, which describes the interaction of  − 
1 0 0 0 
the magnetic impurity and the host (conduction) elec-  
 
trons,shouldbeinvariantunderthecubicgroupsymme- implying
try. Therefore we have to use the Γ symmetry adapted
8
combinations of the host s orbitals (d-type or, more pre- s3/2 = Dx2 y2 ,
| i − − |↓i
cisely, E-type combination of these orbitals). s =D ,
1/2 2z2 x2 y2
| i − − |↑i
The gold host atoms form an fcc lattice: an impurity s = D ,
1/2 2z2 x2 y2
has 12 nearest neighbor host atoms (in the core region | − i − − − |↓i
s =D . (19)
3/2 x2 y2
of a nano-grain). The operators s†xy, s†x¯y, s†xy¯, s†x¯y¯, s†xz, | − i − |↑i
s†yz, s†x¯z, s†y¯z,s†xz¯, s†yz¯, s†x¯z¯, s†y¯z¯ create the appropriate s- By probing for the rest of the generating point group
electrons at the nearest neighbor sites with either spin elements of the cubic point-group it can easily be shown
σ. The subscript of the creation operators refer to the that this basis forms indeed a Γ8 representation of the
nearestneighborsites. Forinstanceiftheimpuritytakes cubic double point-group.
placeatsite(0,0,0)thenindexxydenotesthecoordinate The ionic Hamiltonian, Eq. (13), without kinetic en-
of the site (1,1,0). Using the combinations ergy of the host electrons reduces to
2 2
=E m m +V m 0s +s 0 m ,
1 1 HLSO d | ih | | ih | m †m| ih |
sexy = √2 s†xy+s†x¯y¯ , sex¯y = √2 s†x¯y+s†xy¯ , Xm Xm (cid:0) (2(cid:1)0)
(cid:16) (cid:17) (cid:16) (cid:17)
1 1 where V is the hybridization parameter, and we choose
sexz = √2 s†xz+s†x¯z¯ , sex¯z = √2 s†x¯z+s†xz¯ , E0 =0 and E1,m =Ed for each m. By using a Coqblin-
1 (cid:16) (cid:17) 1 (cid:16) (cid:17) Schrieffer canonical transformation, see Refs. [6] and [5],
seyz = √2 s†yz+s†y¯z¯ , sey¯z = √2 s†y¯z+s†yz¯ , (14) Sec. 1.10,fortheHLSOHamiltonian,Eq. (20),weobtain
(cid:16) (cid:17) (cid:16) (cid:17) that
the d-like combinations, D , can be expressed as
δ HLSO =J s†msm0Xm0m , (21)
mX,m0
1 1
D = (se se ) , D = se se , (15)
xz √2 xz− x¯z yz √2 yz− y¯z where the Hubbard operators, Xm0m =|m0ihm|, refer to
thestates 3,1, 1, 3 oftheimpurity,ands creates
(cid:0) (cid:1) {2 2 −2 −2} †m
appropriatehostelectrons,whileJ denotestheexchange
etc. Including spin variables this leads to ten D σ
δ| i constant,
combinations. The T -type combinations D , D , D
2 xz yz xy
transform as Γ5, while the E-type combinations Dx2 y2 V2
and D transform as Γ . As we mentioned−in J = , (22)
thema2izn2−pxa2p−eyr2, theΓ combinat3ionsarerelevantforthe |Ed|
3
construction of the LSO effective Hamiltonian. To con- where J was set as 0.25 eV (being consistent with the
struct the LSO symmetry adapted quantities, we should Kondo temperature below 0.1 K). This procedure is a
4
natural generalization of the derivation of the s-d model hybridization between the impurity and the s-type con-
by Schrieffer and Wolff for case of j = 1. duction electrons, hence, from the s-components of the
2
We note that the LSO Hamiltonian Eq. (21) is ba- ρˆ (ε) matrix we define the following projected matrix,
C
sically the U = limit of the ionic model. Consid-
ering the large C∞oulomb interaction of the impurity d- ρnn0 (ε)=ρnn0 (ε), (26)
level, the n = 2 double occupation is not allowed and s−C,σσ0 C,sσ,sσ0
theaboveHamiltoniancannotbeobtainedinthissimple where n,n0 and ρˆs is a (2 13) (2 13) matrix,
∈C −C × × ×
form. However, we are convinced that a finite U can re- considering the the up ( ) and down ( ) spin channels.
↑ ↓
sult in a magnetic anisotropy (MA) of the same order of Then we construct a 4 4 ρ∗(ε) matrix from ρˆs re-
× −C
magnitude, i.e. the LSO model is the basic mechanism ferring to its elements by δ1, δ2 and σ, σ0 indices, where
of the partially (but not half) filled magnetic impurities δ1 = x2 y2 and δ2 = 2z2 x2 y2 are the E-type
− − −
embedded into noble metal nanosystems. In Refs. [7] orbitals. NextwetransformthismatrixintotheΓ8 sym-
and [8] Au(Fe) and Cu(Fe) reduced dimensional dilute metry adapted basis by using Eq. (18) as follows,
magnetic alloy systems (thin films) were analyzed, while
Ref. [9] dealt with Cu(Mn) system. In case of Fe impu- ρL(ε)=Qρ∗(ε)Q† . (27)
rities, the suppression of the Kondo effect has been ob-
served but this effect for Mn has not been measured yet.
This observation is in agreement with our expectation
that the LSO model can produce large enough magnetic
anisotropy to explain the reduction of the Kondo effect,
andsuggeststhatthefiniteUdoesnothaveimportance.
APPENDIX C - CALCULATION OF THE a. b.
MAGNETIC ANISOTROPY
FIG. 2: First and second ordered self energy diagrams of the
Here we derive the MA matrix from the LSO model. impurity spin. The dashed and continuous lines denote the
WeshouldnotethatthehostHamiltonian,Eq. (3),must propagatorsofthespinandtheconductionelectrons,respec-
tively.
be modified in the presence of a magnetic impurity that
breaks the two-dimensional translation symmetry. The
Tocalculatethesplittingofthefourstates, weemploy
simplest way to account for this constraint is to shift the
on-site d-state energies of the impurity εi far below the Abrikosov’spseudofermionrepresentation[10]. Usingthe
α
up to second order in J, the diagrams are shown in Fig.
valence band and add the following term to the Hamil-
2, and the self-energy at T =0 temperature is given by
tonian,
∆Hˆ =∆Hαnσ,n,α00σ0 = εiα−εα δn0δn00δαα0δσσ0 , (23) Σmm0(ω =0)=Σ(m1)m0 +Σ(m2)m0 , (28)
where the impurity is a(cid:0)t site n=(cid:1) 0 of a nano-grain. where
According to the LSO model, we need the Green’s εF
functiononlyforaclusterofsites, ,consistingofnearest Σ(1) =J dερL (ε), (29)
C mm0 mm0
neighboratomsaroundtheimpurityandoftheimpurity Z−∞
itself(12+1atoms). ThecorrespondingGreen’sfunction
and
matrix can be evaluated as
gˆ(z)=gˆ0(z) Iˆ−∆Hˆ0gˆ(z) −1 , (24) Σ(m2m) 0 =J2Z−ε∞F dεZεF∞dε0ε01−ερLmm0(ε)Xm00ρLm00m00(ε0),
(cid:16) (cid:17) (30)
where Iˆis a unit matrix and gˆ(z)= Gˆ(z) , and Gˆ(z) whereρL (ε)aretheelementsofρL(ε)computedinthe
is defined by Eq. (10). The sp0ectral{functi}oCn matrix of absencemomf0the exchange interaction, i.e. J = 0, and εF
cluster is then defined as is the Fermi-energy [11]. Interestingly, already the first-
C
ordercontributiontotheself-energygivesanonvanishing
1
ρˆ (ε)= lim(gˆ(ε+iδ) gˆ(ε iδ)) . (25) anisotropy in the vicinity of a surface or at a site of a
C −2πiδ 0 − − nano-grain. Therefore, as what follows we consider this
→
termonly. InthemainpaperweidentifytheMAmatrix
The dimension of the matrices defined in Eqs. (24) and
as the resulted first order self-energy,
(25)is(13 9 2),becausethenumberofsitesinthesmall
× ×
cluster is13,andtheorbitalandspin-indicesare9and
2, respeCctively. Our impurity model is restricted to the {Kmm0}= Σ(m1m) 0 , (31)
n o
5

in the bulk material (the characteristic energy scale of
0
] the MA goesbyN−3/2, see main paper). Thecalculated
V
e-1 valuesinFig. 3areveryfluctuating, whichsuggeststhat
t [m we are far from the bulk behavior. We also note that
tan-2 in case of ordered nano-clusters zero value was obtained
s for the central atom in agreement with the theoretical
n
o-3
c investigations based on symmetry analysis.
y
op-4 TheparametersoftheMAmatrixcanbecalculatedfor
tr eachcoresiteinagivennano-grain. Thematrixelements
o
is-5 aredifferentfordifferentsiteseveninthesameshell,but
n
a symmetry relations were found between the sets of the
tic-6 MAparameters. Letusenumeratethecoresitesbyindex
e
n
g i=1...87. We label expression (32) by a site index i, i.e.
a-7
M 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 we now introduce a set of the HiL-s. The parameter set
K will have a site index i, too. Let i and j be two
S ite in d e x o f th e n a n o c lu s te r { µ}
sites in the same shell and let the two sites be connected
by the group element g O where gR = R . So the
h i j
∈
FIG.3: (coloronline)CalculatedMAconstant,K ,valuesfor transformation rule for the matrices HL and HL can be
L i j
both ordered and disordered nano-balls. The ordered cluster written as
hasN=225atoms,thesolidlinescorrespondtothevaluesof
KL calculatedfortheorderedcase. Thenumberofcoresites HjL =eiϕn·JHiLe−iϕn·J , (33)
is N =87, i.e. the MA is for atoms located in the center site
c
and on the first six core shells, see the different colors. The where J = (Jx,Jy,Jz) is the angular momentum-vector
numberofallsitesindisorderednano-ballsisN=265,i.e. 40 operator,whileg correspondstotherotationaroundaxis
extra atoms are put to the next three outmost shells. The n with angle φ. From Eq. (33) we can derive relations
number of sample is NS =50. between the anisotropy matrixelements of the different
sites.
Motivated by the quadrupole-decomposition of the
and the effective spin Hamiltonian can be written as
anisotropy matrices, it seemed reasonable to calculate
HL = K m m . (32) the transformation rules for the coefficients in Eq. (5)
mm0| 0ih | of the main paper. The quadrupolar operators form the
mX,m0
basis of a real five-dimensional vector space (parameter
The structure of HL is given by Eq. (3) in the space), and the transformation matrices between the pa-
main paper, the MA matrix can be parametrized by rameter sets of the different sites i and j are defined as
five real numbers, K (µ = 1...5). The difference of
µ Kj =Γ(i j)(g)Ki , (34)
itsKramers-degenerateeigenvaluesdefinestheLSOmag- →
netic anisotropy constant K (see Eq. (4) in the main
L where Ki(j) is a vector formed by the set of Ki(j) .
paper). The MA constant should be the same for sites µ
in the same shell because of the Oh symmetry relations. The transformation matrices Γ(i→j)(g) have a vnery spoe-
The thick solid lines in Fig. 3 show the corresponding cial structure, namely they are block-diagonal with two-
MAvaluesfordifferent(core)shellsincaseofanordered and three-dimensional blocks: Eg(i→j)(g) and T2(gi→j)(g)
nano-grain with 225 atoms. If one randomly puts 40 ex- in order. The five-dimensional parameter space is de-
tra atoms on the sites of the next three outmost shells, composed into two (dim=2+3) subspaces.
constructing disordered nano-grains, then the MA is not Whereas the two-dimensional E parameter subspace
shell-degenerate anymore. has been discussed in the main paper, we focus here just
It should be also noted that in case of tetragonal sym- onthethree-dimensionalT -space. WenotethattheK ,
2 3
metry, e.g. whenthemagneticimpurityisinthevicinity K andK parametersspantheT spacebeingbyabout
4 5 2
of a surface of a film or bulk material, we obtain that one order smaller in magnitude then the parameters on
theMAhasonlydiagonalnon-zeroelements, i.e. theen- theE-plane. Fig. 4showsthecalculatedMAparameters
ergy difference of the 3/2 and 1/2 states defines the MA intheT spacefortheorderednano-grainwith225atoms
2
constant (K = K ). Moreover, in case of cubic sym- (87coreatoms). Wecanidentifythetetrahedralordering
L 1
metry, when the magnetic impurity takes place in the pattern of the parameters in agreement of the structure
bulk, K =0 and, therefore, K =0 as well in agreement oftheT transformationmatrices. Theorderedstructure
1 L 2
with the statement that the D3/2 ground state (Γ ) re- can be split even putting only one extra atom to the
8
mains degenerate in a cubic crystal field. This implies outmost shell of the spherical cluster.
that the magnetic anisotropy should go to zero at the SimilarlytothecaseofE-plane,weexaminetheradial
inner shells when increasing the size of nano-grains like distribution of the MA parameters in the T parameter
2