Temperature dependence of the magnetic hyperfine field at an s-p impurity diluted in R Ni 2
A. L. de Oliveira, C. M. Chaves, N. A. de Oliveira, A. Troper
TTemperature dependence of the magnetic hyperfine field at an s-p impurity diluted in R Ni A. L. de Oliveira ∗ Instituto Federal de Educa¸c˜ao, Ciˆencia e Tecnologia do Rio de Janeiro,Campus Nil´opolis, Rua L´ucio Tavares 1045, 26530-060, Nil´opolis, RJ, Brazil
C. M. Chaves † and A. Troper Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil
N. A. de Oliveira
Instituto de F´ısica Armando Dias Tavares, Universidade do Estado do Rio de Janeiro,Rua S˜ao Francisco Xavier 524, 20550-013, Rio de Janeiro, Brazil
We study the formation of local magnetic moments and magnetic hyperfine fields at an s-p im-purity diluted in intermetallic Laves phases compounds R Ni ( R = Nd, Sm, Gd, Tb, Dy) at finitetemperatures. We start with a clean host and later the impurity is introduced. The host hastwo-coupled ( R and Ni) sublattice Hubbard Hamiltonians but the Ni sublattice can be disregardedbecause its d band, being full, is magnetically ineffective. Also, the effect of the electrons of R isrepresented by a polarization of the d band that would be produced by the magnetic field.Thisleaves us with a lattice of effective rare earth R -ions with only d electrons. For the dd electronicinteraction we use the Hubbard-Stratonovich identity in a functional integral approach in the staticsaddle point approximation. I. INTRODUCTION
The Laves phases intermetallic compounds R Ni ( R =rare earth elements ) crystallize in a cubic structure [1]and exhibit an interesting variety of behaviors relatedto the changes in their magnetic, electronic, and latticestructures. They exhibit magnetization associated bothwith the localized spins (4 f ) and with the itinerant elec-trons of the rare earth. Although the extensive studiesmade, the theoretical description of how some of theirproperties varies with temperature, due to the electroniccorrelations, still remains open. The Ni sublattice how-ever can be disregarded because its d band, being full,is magnetically ineffective. Also the effect of the R electrons is represented by a polarized d band producedby the electrons magnetic field. This approximationgenerates an effective rare earth R lattice but whose in-teractions differ from the usual R pure metal because nowthe equivalent lattice constant is different (see the end ofsection III for a numerical indication of the differencebetween the two cases).
II. METHODII.1. The effective host
We start with a clean effective host described by theHamiltonian of itinerant d -electrons: H = H d + H d . (1) ∗ [email protected] † [email protected] The first term of Eq.(1) is H d = (cid:88) lσ ε σ d † lσ d lσ + (cid:88) ll (cid:48) σ T ll (cid:48) d † lσ d l (cid:48) σ , (2)where ε σ is the energy of the center of the d band, nowdepending on the spin polarization; d † jσ ( d jσ ) is creation(annihilation) operators, and T ll (cid:48) is the hopping integralsbetween atoms from the R effective lattice. H d represents the Coulomb interaction H d = U (cid:88) l n l ↑ n l ↓ , (3) n lσ being the number operator.The partition function Z for the system described byHamiltonian (1) can be written as Z = (cid:90) (cid:89) l d ν l d ξ l (cid:90) (cid:89) j d ν j d ξ j e − β [ − U (cid:80) l ( ν l + ξ l )] × tr e − β ( H d − U (cid:80) lσ ( iν l + σξ l ) n lσ ) · (4)where β = 1 /k B T , k B is the Boltzman constant and T thetemperature. The Hubbard-Stratonovich identity[2] wasused in order to linearize the Coulomb interaction gener-ating two floating fields, an electric, ν l , and a magnetic, ξ l . The static approximation has also been performed.Now, because of the floating fields, the system, al-though pure, becomes disordered[3, 4]. In Eq.(4) we seethat the floating fields create site dependent ε l σ ’s : ε l σ = ε σ − U iν l + σξ l ) (5)We then adopt the Coherent PotentialApproximation[5, 6] (CPA) point of view in which a r X i v : . [ c ond - m a t . s t r- e l ] A ug the system is replaced by an ordered one with anuniform self energy Σ σ in all sites l (cid:54) = l ; in l theenergy ε l σ remains function of the fluctuation fieldswith respect to Σ σ . The partition function in Eq (4) isalso the partition function for the Hamiltonian below(which will be the one used to implement the CPA)˜ H = ˜ H + ˜ H , (6)with˜ H = (cid:88) σ [ ε l σ − Σ σ ] d † l σ d l σ = (cid:88) σ V σ d † l σ d l σ , (7)and ˜ H = (cid:88) lσ Σ σ d † lσ d lσ + (cid:88) ll (cid:48) σ T ll (cid:48) d † lσ d l (cid:48) σ (8)In (8) both l and l (cid:48) are (cid:54) = l . Summing over all possible l we arrive at the self-consistency condition to determineΣ σ : (cid:90) d ξ l d ν l V σ ( ξ l ν l )1 − V σ ( ξ l ν l ) g l l σ ( z ) P ( ξ l ν l ) = 0 . (9)where the probability distribution P , is P ( ξ l ν l ) = e − β Ψ( ξ l ν l ) (cid:82) d ξ l d ν l e − β Ψ( ξ l ν l ) , (10)and Ψ is the free energy associated to ˜ H ,Ψ( ξ l ν l ) = U (cid:0) ξ l + ν l (cid:1) + 1 π (cid:90) d ε f ( ε ) Im (cid:88) σ ln [1 − V σ ( ξ l ν l ) g l l σ ( z )] , (11) f ( ε ) is the Fermi function, z = ε + iδ , δ −→ + , and g l l σ ( z ) is the Green function for the hamiltonian ˜ H .Then, the partition function (4), in the CPA approachis reduced to Z = (cid:90) (cid:89) l d ν l d ξ l e − β Ψ ( ν l ,ξ l ) . (12) II.2. The introduction of a s-p impurity
We now describe the effects caused by the introductionof a s-p impurity (say Cd). We add a potential V imp0 σ toEquation 6, V imp0 σ = (cid:2) ε imp σ − Σ σ ( z ) (cid:3) d † l σ d l σ , (13)where the impurity energy ε imp σ is self consistently de-termined using the Friedel [8] condition for the chargesscreening ∆ Z = ∆ Z ↑ + ∆ Z ↓ , (14) FIG. 1. Local magnetic moment at Cd impurity diluted in R Ni intermetallic host for light rare earths. where ∆ Z σ is the total charge difference between the σ conduction electrons of the impurity and the host:∆ Z σ = ln (cid:8) − g l l σ ( (cid:15) F ) (cid:2) ε imp σ − Σ σ ( (cid:15) F ) (cid:3)(cid:9) . (15)Using the Dyson equation, the perturbed Green func-tions for this problem can be written as G l l σ ( z ) = g l l σ ( z )1 − g l l σ ( z ) (cid:104) ε imp σ − Σ σ ( z ) (cid:105) . (16)The local density of states for the σ spin direction is ρ σ ( ε ) = − π Im G l l σ ( z ) (17)and the local occupation number is n σ = (cid:90) (cid:15) F −∞ ρ σ ( ε ) f ( ε ) d ε. (18)So, the magnetic moment at the impurity site is (cid:101) m (0) = (cid:88) σ σn σ (19)and finally, we calculate the magnetic hyperfine field atthe impurity site, assuming that it is proportional to (cid:101) m (0), via the temperature independent A ( Z imp ) Fermi-Segr`e contact coupling parameter [7].: B hf = A ( Z imp ) (cid:101) m (0) , (20) III. RESULTS AND DISCUSSIONS
We have introduced an effective model to extend ourprevious zero temperature results [8] to investigate the
FIG. 2. Local magnetic moment at Cd impurity diluted in R Ni intermetallic host for heavy rare earths. magnetic hyperfine fields at a s-p impurity such as Cd,in R Ni at finite temperatures. We adopt a standard paramagnetic density of state extracted from first princi-ple calculation [9]. For each R Ni compound our modelhas two adjustable parameters, namely ε σ and U . Theseare determined by reproducing the zero temperature lo-cal magnetic moment and the critical temperature.In Fig. 1 , Fig. 2 and Fig. 3 we plot the calculatedtemperature dependence of the local magnetic momentsfor the light rare earth elements, for the heavy elementsand the magnetic hyperfine fields at Cd as function oftemperature.The results are good agreement with the ex-perimental results [10].As stated before, in the effective R lattice the inter-actions are different from a R pure metal. From Fig. 2we see that the local moment, in units of Bohr magneton µ B , at T = 0 K is about 0.012 whereas in Ref. 11 it wasfound that in pure Gd metal it is about 0.05. ACKNOWLEDGMENTS
We would like to aknowledge the support from theBrazilian agencies FAPERJ and CNPq. [1] K.H.J. Buschow. Rep. Prog. Phys., , 1179, (1977).[2] J. Hubbard, Phys. Rev. Lett. ,77 (1959).[3] H. Hasegawa, J. Phys. Soc. Japan , 1504 (1979).[4] Y. Kakehashi, Phys. Rev B , 9207 (1990).[5] B. Velick´y, S. Kirkpatrick, and H. Ehrenreich, Phys. Rev. , 747 (1968).[6] H. Hasegawa, J. Phys. Soc. Jpn. 49, 963 (1980).[7] I.A. Campbell, J. Phys. C , 1338 (1969). [8] A.L. de Oliveira, N.A. de Oliveira and A. Troper, Phys.Rev. B , 12411 (2003).[9] H. Yamada, J. Inoue, K. Terao, S. Kanda and M. Shim-itzu, J. Phys. F: Met. Phys. , 1943 (1984)[10] S. M¨uller, P. de la Presa and M. Forker, Hyperfine In-terac. , 163 (2004).[11] A. L. de Oliveira, M. V. Tovar Costa, N. A. de Oliveiraand A. Troper, J. Appl. Phys. ,4215 (1997). D y N i T b N i G d N i N d N i S m N i Magnetic hyperfine fields (Tesla)
T e m p e r a t u r e ( K e l v i n )
FIG. 3. Magnetic hyperfine field at Cd impurity diluted in R Ni2