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A simple description of the low-temperature magnetic properties of (Zr1−xHfx)Fe2 pseudobinaries
Journal of Magnetism and Magnetic Materials 114 (1992) 283-290 North-Holland
A simple description of the low-temperature of ( Zr, _-xHf .) Fe, pseudobinaries N.A. de Oliveira
and A.A. Gomes
magnetic properties
’
Centro Brasileiro de Pesquisas Fisicas (CBPF-CNPq),
Rua Dr. Xavier Sigaud 150, Urea, Rio de Janeiro, RJ 22290, Brazil
Received 9 October 1991
The possibility of ferrimagnetism and the concentration dependence of the magnetization in (Zr, _,Hf,)Fe, compounds are discussed within a parameterized TB-CPA calculation at low temperatures. The good agreement with available. data for low temperatures and its computational simplicity make the approach suitable for future treatment of temperature effects.
1. Introduction The connection between the magnetic properties and the electronic structure of Laves phase pure and pseudobinary intermetallics is an active subject of both experimental [ll and theoretical research [2]. First-principle calculations have been performed in pure intermetallics and ordered compounds to simulate some pseudobinary intermetallics [2]. The case of pseudobinary compounds, including the detailed effect of substitutional disorder, is much more complex. A first attempt to describe (Zr,_,Hf,)Fe, within a very simplified model appeared some time ago [3]. In that work, the existence of two sublattices was considered only as providing a way to estimate the charge transfer, the Fe sublattice acting as a reservoir. The absence in ref. [3] of interaction between the sublattices inhibited the possibility of the disorder in the A sublattice to modify the local 3d density of states. In the present work we suggest an intermediate description staying between the first-principles approach and the previous oversimplified Correspondence to: Dr. N.A. de Ohveira, Centro Brasileiro de
Pesquisas Ffsicas (CBPF-CNPq), Rua Dr. Xavier Sigaud 150, Urea, Rio de Janeiro, RJ 22290, Brazil. ’ IF-UFRGS/CBPF-CNPq. 0304-8853/92/$05.00
model [3]. Electronic structure models usually are based on a tight binding (TB) description of the electronic states, recently revived by the use of the LMTO formalism [4]. This is a parameter free theory, but as mentioned before, the case of pseudobinqries with several atoms per cell and including disorder requires a larger computational effort and some technical difficulties to be overcome. We thus decided to adopt a simple two-sublattice model [51 and a convenient parametrization scheme to deal with the following points determining the magnetic properties: (a) Substitutional disorder in one of the sublattices, as described within the coherent potential approximation (CPA), and its role in modifying the charge transfer between the sublattices. (b) The relation between the ferrimagnetism of these materials and the interactions between sublattices. In the following we present the model, the parametrization procedure, and an application to the magnetic properties of (Zr,_,Hf,)Fe, pseudobinaries. Emphasis is given on the computational simplicity of the model and its capability of accounting for the magnetic data. The model may be tested using the formalism [6] proposed by Heine et al. for magnetostriction phenomena, when experimental data will be available. Also, this calculation is used as the starting point for
0 1992 - Elsevier Science Publishers B.V. All rights reserved
N.A. de Oliveira, A.A. Gomes / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
284
the finite-temperature description of these compounds via the functional integral formalism [7].
2. Formulation The pseudobinary compound (Zr 1_-xHf x)Fe, contains two interpenetrating sublattices, and Hf impurities are selectively introduced only in the A sublattice of the AB, compound. We describe this system using the TB formalism within the simplifying approximation of five identical local d-subbands. Let us define the corresponding Hamiltonian:
ULni f ni 1 + C $%,~aj,
+ C jj’a
TJFa,~ajvu + C U~nj t nj,
+ C
~~(UifolZic,
j U~Ujm),
eikRii.
(2c)
(1)
ijo
i = Zr or Hf,
E; = c; + U&r_,),
where a,‘, (aj,) is the creation (destruction) operator of one electron at site “9 (“j”) with spin u in the A (B) sublattice. The energy of a d-level at site i of the A sublattice, depending on the occupancy of this site, can take values E? or co”‘. In the B sublattice it can take only the value l r, since impurities are selectively dissolved in these materials. The terms Tiy (Tjy) represent hoppings between “ii”’ (“jj”‘) within the same A (B) sublattice, TF are hoppings between the A and B sublattices. Finally, the Coulomb interaction strengths are introduced in the Hamiltonian through the parameters Up, UAHfand Up. Some approximations are introduced now. (a) Firstly, since Hf and Zr are 5d and 4d elements, respectively, one has different local bandwidths, and thus off-diagonal disorder. To include in the simplest way off-diagonal disorder, an x-dependent bandwidth, given by the virtual crystal approximation may be adopted: A/,(X) = (1 -x>
q;:4” = CT,
between the corresponding sites and the k-dependent functions E:, E; and rk will be defined below. cc> The Coulomb interaction is dealed with using the Hartree-Fock approximation. This reduces the problem to define effective spin dependent energy levels of Hf, Zr and Fe through:
iv
+
P)
Riir, Rjjr and Rij are the separation
ii’a
+ C i
(2a)
k
H = ~&zi+,aim + c Ti~ai+,ait, ia
comparable bandwidths, as will be estimated below. The local bands associated with Zr and Hf atoms located at the A sublattice and centered at energies EF and crf, are then taken to have a common width A,(x). (b) The second approximation concerns a model description of the two-sublattice system, defined by [5]:
eFe 0” + UFe(nj_u). I7 = l
(3b)
The substitutional disorder introduced in the A sublattice is described within the CPA approximation. The levels EL in eq. (3a) are replaced by an effective medium J$,(z>, with z = E + ia, 6 -+ O+. This effective medium is determined self-consistently as follows: within the CPA description the Hamiltonian for an A sublattice configuration of effective atoms, except a given Zr or Hf atom occupying the origin, taken in the A sublattice is: H=H,+V*,
(4a)
where ia + C
ii ‘17
e,Fea,Lajo + C Tj~al~aj~q
(4b)
A,, +x Am,
where the A’s are Zr and Hf local bandwidths. This is reasonable since both Zr and Hf have
(3a)
VA=
(4c)
N.A. de Oliueira,AA. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x HfJFe,
The Green function (or resolvent) for the system corresponding to the Hamiltonian (4) is defined by a (2 x 2) matrix, reflecting the two-sublattice character of the material:
G=($I“,::)y G=g+gVG,
The resolvent becomes then: 2
-&(z) %-YEk
(64 from which one obtains performing braic operations:
w
Ly(1 -$/a)(E”,-E”) X[(Z-•:-E?)F(K)
( 1 VA 0 0
some alge-
1 F*a(Z)=gi~=
where the potential is given by the matrix: v=
--‘?I(
(5a)
and should satisfy the matrix Dyson equation:
285
0
-(z-e:-EO,)F(E”,)],
and g = (z - H,)- ’ is the unperturbed (2 X 2) matrix Green function for the Hamiltonian defined in terms of the effective medium. Using eq. (2) the unperturbed Green function in reciprocal space can be simply writen as:
(6d) 1 F,B(z)=g;;=
(Y(1 - y2/(Y)( E”,-E”_)
x[(z-i&,-aE’T)F(EI) -(z-Zo.-aE”,)F(E”,)], (he)
(6a) In order to simplify the numerical calculation of the unperturbed resolvent, let us introduce the following model for the two-sublattice system (2):
where 1 E”,= 2a(l -$/(Y) x
(1) E;=Q, (2) Et = ffEk) (3) r, =r,
[z-z,-2r,y+++)] t
+--z,+2r,y++-~,Fe)]2
(6b)
+ YEk,
where lk is a dispersion relation defined later on. In order to understand the physical meaning of the above approximations, let us recall the principles of an LMTO-TB calculation. Two different ingredients appear in a two-sublattice system: first, one must introduce structure functions associated, respectively, to the A and B sublattices; these structure functions reflect the geometric characteristics of the intermetallic. Second, three potential parameters [4] are associated with the involved atoms Fe, Zr and Hf. Thus the main advantage of this homothetic band model is to replace the structure functions by a single one and to incorporate in the parameters (Y, r, and y, the intra- and inter-sublattice hopping strength.
-4[(~-Z,)(~-~,Fe)-r~](~~2)}~"} (69 and F(z)
= /p(
l)/(
z-
l) de.
(6g)
The results (6), being analytical expressions for the local Green functions, simplify the numerical calculations, in particular for AB, systems with six atoms per unit cell. The general solution of Dyson’s equation (5b) in real space is:
v-a)
286
N.A. de Oliueira,A.A. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x HfJFe,
The local Green function for an atom i = Zr or Hf embedded in the effective medium follows from eq. (7a):
cY4
Gzw = 1 -
Using the standard procedure, is obtained from:
(‘-‘) 1-
(7b)
(EL -&(z))F,A(z)
the self-energy X,,
P-&(Z)
mre = 5( n: - .?J,
( 1Ob)
the factor 5 accounting for the degeneracy of the d-band. The magnetic moment per formula unit of the pseudobinary is given by: +xm,,
+ 2mFe.
( 1Oc)
3. Results of the numerical procedure Since the pseudobinary (Zr,_,Hf,)Fe, involves substitution of isoelectronic elements in the A sublattice, the total number of electrons should be independent of the concentration. Consequently any change in the average occupation number of the A sublattice will imply in a charge transfer between the A and B sublattices. The total number of electrons for a Laves compound is given by:
EF
/
Piu(E)
nF= = EFP,Fe (e) (T /
de, de,
(9a)
n,=&(X)
(9b)
where the occupation number of the A sites is given as a function ot the concentration x: by:
where or is the Fermi level and the local densities of states are:
piv(e)
( 1Oa)
[&Jz)]F,A(z)
Formally this is a quite similar result to that corresponding to the usual description of simple alloys; the important difference is contained in eqs. (6) for the functions F/(z) and F:(z 1, where the coupling between sublattices is included. This extends the early description [3], because disorder now dktorts also the B sublattice density of states. The number of electrons is obtained from:
pp=
the
mi = 5( IZ~~- IZ~_~), i = Zr or Hf,
m = (1 -x)m,,
(8)
ni, =
precision window. After self-consistency, magnetic moments are calculated from:
GTrn = GIrn
F:(z), G,4p,“‘( z),
(SC) i=
Zr or Hf.
(94 Eqs. (8), (7b), (91, (6), and (3) must be solved self-consistently. The procedure by which these coupled equations are solved is the usual one: given an initial set of parameters (Y, y, r,, U?, UAHf,and UFe, the dispersion relation Ed and values for the occupation numbers nk, using these equations one obtains an initial value for the self-energy 2,. One then recalculates the corresponding occupation numbers. This process is repeated until self-consistency, within a 0.002
n:(x)
+2&(x),
= (1 -.x)+,(x)
( lla)
+xn,,(x),
(lib)
and the occupation numbers n,(x), i = Zr, Hf or Fe are given by eqs. (9). We decided to start our fitting procedure from the x = 0 limit of the series because: (i) The density of states P(E) for the Fe sublattice to be used in eq. (6g) is obtained from the ZrFe, calculation of Terao and Shimizu [8], for the Cl5 structure, keeping only the main features and normalizing to unity (see fig. 1 for a comparison). (ii) The occupation number at the Fe site has been also extracted from the ZrFe, calculation
Bl. (iii) Finally and more importantly, it is well known that HfFe, shows coexistence of the Cl4 and Cl5 phases, which complicates the procedure of the fit.
N.A. de Oliveira,A.A. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x Hf,)Fe,
Fig. 1. Local 3d density of states as calculated in ref. [S] (dashed line) and simplified one (full line).
Since we use the homothetic model, eq. (6b), in performing the numerical calculations we need an alternative estimate for n.,(O) = n$‘i and in the opposite concentration limit, n,,(l) = n$& It should be remembered that compounds of Hf and Zr are isoelectronic, and this should be ensured during the calculation for any value of x. To estimate the number of d-electrons per site for the specific sites occupied by Zr or Hf atoms in the extreme concentration limits, we adopt the following procedure, as dictated by recent literature [2]. We approximate the local bands by constant and normalized densities of states, their centers and widths being estimated from the parameters of the self-consistent LMTO-ASA calculations tabulated in ref. [4] for the pure elements. These are given by:
lre = 0 7
A,, = 2,
lZr = 1.39,
A,, = 4.4,
EHf = 1.77,
A,, = 5.1.
Converting these quantities to reduced units where the Fe half bandwidth is taken equal to 1, and assuming that the number of electrons at the Fe site in the pure Cl5 compound is given by ref. [81 ($‘L = 1.44 = (7.2/5)), one gets er = 0.44, in our units. This choice implies in the following electron occupation numbers at the Hf and Zr sites: n$$ = 0.48 and $‘i = 0.56. Note that the total and concentration independent number of
287
electrons nT is obtained from eq. (111, using these values for x = 0. Our estimate of these occupation numbers indicates that charge transfer should occur between the A and B sublattices in order to restore the isoelectronic character. This transfer is automatically included in the formalism through eqs. (11) and (9); our estimate of the occupation numbers is only needed in order to fix, in the paramagnetic phase, the values of E? and ~0”’ for the compound, given the Coulomb interaction strengths. Within the description in terms of dominating diagonal disorder, the bandwidth of the local bands of Zr and Hf are taken equal. Thus the dispersion relation for the A sublattice in the homothetic model, eq. (6b), is proportional to that of the Fe sublattice, with average scaling factor (Y= 2.2. If a more rigorous approach is needed, a scaling factor which is concentration dependent due to the virtual cristal approximation, should be included. The Coulomb interaction parameters are chosen in order to reproduce the experimental magnetic moments of the pure intermetallic ZrFe,. The results for these parameters in our units are: Ur’ = 1.5 and Uz’ = 1.0; for Hf we have assumed UHf = 0.7. These values are consistent with the increase in extent of the d-wave functions in passing from Fe to Hf. Using these values of the Coulomb interaction parameters and the occupation numbers, the energy levels ~5 in eq. (3) may be fixed for decoupled lattices. We obtain E? = 1.963 and l f’= 2.198 and the so obtained local densities of states are shown in figs. 2 and 3. Now we turn on the inter-sublattice coupling, approximated in eq. (6b) by a constant hybridization value r, (see ref. [53> plus a k-dependent term. The constant term r,, corresponds to the usually adopted hybridization in model Hamiltonians. The range of inter-sublattice hopping in eqs. (2~) and (6b) is controlled by the k-dependence of lk; the parameter y in eq. (6b) defines the relative importance of the “hopping” between the A and B sublattices and the constant hybridization term. It has been shown by Yamada and Shimizu [9] that the hybridization between the spin polarized
288
N.A. de Oliueira, A.A. Games / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
150\
DENSITY
OF STATES
L--_ r3.0-
- - -4%
> 5.0 ENERGY
1.50 f
Fig. 2. Local densities of states for ZrFe* in the paramagnetic phase. Local 3d density of states (full line), local 4d (dashed line). The position of the Fermi level is indicated by the vertical line.
3d-states and the higher-energy states associated with the 4d or 5d atoms plays the central role in explaining the occurrence of ferrimagnetism in these compounds. Since hybridization in their paper may be interpreted as transferring electrons mostly confined to one of the sublattices to the other, we hope to reproduce the results of ref. [9] by suitably parametrizing the r, and y values. In ref. [9] the splitting of the up and down 3d-states implies that admixture, at the A sublattice sites, of the down spin states is larger than for up spin ones. Thus, the A sublattice down spin occupation number is expected to be larger than for the up spin case. These remarks suggest to allow for spin dependent values of the parameters r, and y, which we refer to as r,,r, roL and y f, y 1. We considered two extreme limits in order to obtain ferrimagnetic solutions r,‘/r,,l f 0 and y ?/y L = 0 or r,,T/rO1 = 0 and y T/y 1 # 0. These are the central parameters to establish ferrimagnetism through the unbalance of the interactions of different spins. The used values in this calculation are the following: r,f = 0.1, r,l = 0.9 or y T = 0.7 and y 1 = 0.2.
These particular choices of the parameter ratio can be roughly understood as follows. The k-independent parameter mixes states associated with the A and B sublattices, independently or real space considerations. If one intends to increase the occupation number of a given spin state, the more states are admixed, the larger the occupation number. On the contrary, according to eqs. (2~) and (6b), for a given k-dependence of lk, the larger the value of the parameter y, the larger the probability of a given spin electron to hop out, off the original A sublattice. The reverse hopping, namely from B to A, is in a way combined with the effect of a larger 3d Coulomb interaction in the B sublattice, which tends to conserve the difference in spin occupation numbers at B, which could be altered by the inter-sublattice hopping. To conclude, there is an effective inter-sublattice hopping which is larger from A to B and prefers up spin electrons. Thus at the end of the process there is smaller occupation of up than down spins at the A sublattice, thus ferrimagnetic behavior. In order to numerically verify how the relative strengths of the two described processes affect the existence of ferrimagnetic properties of the
DENSITY
OF STATES
Fig. 3. Local densities of states for HfFe, in the paramagnetic phase. Local 3d density of states (full line), local 5d (dashed line). The position of the Fermi level is indicated by the vertical line.
289
N.A. de Oliveira, A.A. Comes / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
Table 1 Magnetic moments as a function of the coupling strengths
compound, we have performed calculations using a sort of linearly interpolated parameters defined as:
P
r&
ror
YT
Y1
mz,
mFe
or mHf
Zr compounds 0.10 0.01 0.09 0.1 0.08 0.2 0.07 0.3 0.06 0.4 0.05 0.5 0.04 0.6 0.03 0.7 0.02 0.8 0.01 0.9 0.00 1.0
where F{(p) = (1 - p)T,” and y”(p) = py”. The two opposite limits are thus recovered, and intermediary values are thus obtained, which provides the crossover between the processes. The results of this calculation are shown in table 1. In this table one sees how critical the strengths of the two processes are in passing from ferrimagnetism at the ends of the range of p to ferromagnetism for intermediate regions. The numerical results show that for smaller values of the coupling parameters ferrimagnetic solutions are unstable. To verify this statement we calculated two “pure” intermediate situations: (a) TOT= 0.05, raL = 0.45 or y t = 0.0 and y J = 0.0 and (b) rof = 0.0, r’,l = 0.0 or y T = 0.42 and y L = 0.12. Case (a) is weakly ferromagnetic, whereas case (b) is weakly ferrimagnetic. These are values separating stability regions. In fig. 4 the variation of the magnetic moments in the A sublattice is shown as a function of the concentration. Note that the Zr and Hf magnetic moments at the A sites are always antiparallel fo the Fe magnetic moment. This calculation was performed using pure hopping-like interactions.
-I 4.00
Hf compounds 0.10 0.0 0.09 0.1 0.08 0.2 0.07 0.3 0.06 0.4 0.05 0.5 0.04 0.6 0.03 0.7 0.02 0.8 0.01 0.9 1.0 0.00
0.90 0.81 0.72 0.63 0.54 0.45 0.36 0.27 0.18 0.09 0.00
0.00 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.70
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
+ 0.49 - 0.28 - 0.08 0.08 0.20 0.24 0.19 0.03 -0.10 - 0.34 - 0.50
1.67 1.75 1.87 1.98 2.10 2.18 2.20 2.10 2.03 1.98 1.90
0.90
0.00
0.00
0.81 0.72 0.63 0.54 0.45 0.36 0.27 0.18 0.09 0.00
0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.70
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
- 0.52 - 0.33 -0.16 - 0.03 0.06 0.14 0.07 -0.00 -0.16 -0.30 - 0.41
1.50 1.57 1.65 1.73 1.94 1.94 1.97 1.99 1.87 1.76 1.69
In fig. 4 the average magnetic moment of the sublattices and the total magnetization of the
f ’ + ’ + e,
-m 5g 3.00 2.00 I
q
0
0
Q
0
cl
$ 0 t;
z
2
1.00
t
0.00 .
0
0..
I
I
I
I
0.2
0.4
06
0.8
.o
l
0
0
0
I
l
1.0 CONCENTRATKH l mn
X1
-1.ooL Fig. 4. Magnetic moment of the pseudobinary compound (in Bohr magnetons kb): mA and 2mFe are the magnetizations at the A and B sublattices, respectively.
N.A. de Oliveira, A.A. Games / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
290
Table 2 Magnetic moments as a function of concentration X
“2
0.0
- 0.505 -0.510 -0.510 - 0.520 - 0.520 - 0.535 - 0.550 - 0.550 - 0.555 - 0.560 - 0.570
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mHf
- 0.383 - 0.385 - 0.385 - 0.390 - 0.390 - 0.395 - 0.400 - 0.400 - 0.405 -0.410 - 0.415
mA
-
0.505 0.497 0.485 0.481 0.468 0.465 0.460 0.445 0.434 0.425 0.415
parameters as used in the construction are used here.
mFe
mAB2
1.900 1.875 1.840 1.830 1.805 1.790 1.750 1.735 1.725 1.710 1.690
3.295 3.253 3.195 3.179 3.142 3.115 3.040 3.025 3.016 2.995 2.965
pseudobinary are shown, the results being indicated in table 2. In fig. 5 the spin polarized density of states is shown for the concentration x = 0.5. This figure illustrates the effect of the spin-dependent hopping in producing an antiparallel magnetic moment at the A sites. The same
of fig. 4
4. Conclusions Using a parameterized calculation it was possible to account for the magnetic properties of (Zr, _,Hf,)Fe, compounds. The advantage of this parametrization was that it clearly exhibits the relevance of each parameter for the concentration dependence of magnetic moments and more importantly for the occurrence of ferrimagnetim; thus this may serve as a guide for more precise and detailed ab initio calculations. Also, such a calculation enables the functional integral description of the temperature dependence of the magnetizations, which becomes simpler if the one-electron densities of states are extracted from this parametrized version.
Acknowledgements 1.50
DENSITY
OF STATES
The authors would like to thank C. da Silva for discussions and a critical reading of the manuscript.
References
1.501 ” Fig. 5. Spin-polarized local densities of states for the concentrations x = 0.5. The position of the Fermi level is indicated by the vertical line.
[l] F. Baudelet et al., Europhys. Lett. 13 (1990) 751. M. Iijima et al., J. Phys.: Condens. Matter 2 (1990) 10069. [2] 0. Eriksson et al., Phys. Rev. B 40 (1989) 9519. [3] L. Amaral, F. Livi and A.A. Comes, J. Phys. F 12 (1982) 2213. [4] O.K. Andersen et al., in: Highlights in Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (NorthHolland, Amsterdam, 1985). [5] J. Giner et al., J. Phys. F 6 (1976) 1281. [6] V. Heine et al., J. Magn. Magn. Mater. 43 (1984) 61. [7] N.A. de Oliveira and A.A. Comes, to be published. [8] K. Terao and M. Shimizu, Phys. Stat. Sol. (b) 139 (1987)
A simple description of the low-temperature of ( Zr, _-xHf .) Fe, pseudobinaries N.A. de Oliveira
and A.A. Gomes
magnetic properties
’
Centro Brasileiro de Pesquisas Fisicas (CBPF-CNPq),
Rua Dr. Xavier Sigaud 150, Urea, Rio de Janeiro, RJ 22290, Brazil
Received 9 October 1991
The possibility of ferrimagnetism and the concentration dependence of the magnetization in (Zr, _,Hf,)Fe, compounds are discussed within a parameterized TB-CPA calculation at low temperatures. The good agreement with available. data for low temperatures and its computational simplicity make the approach suitable for future treatment of temperature effects.
1. Introduction The connection between the magnetic properties and the electronic structure of Laves phase pure and pseudobinary intermetallics is an active subject of both experimental [ll and theoretical research [2]. First-principle calculations have been performed in pure intermetallics and ordered compounds to simulate some pseudobinary intermetallics [2]. The case of pseudobinary compounds, including the detailed effect of substitutional disorder, is much more complex. A first attempt to describe (Zr,_,Hf,)Fe, within a very simplified model appeared some time ago [3]. In that work, the existence of two sublattices was considered only as providing a way to estimate the charge transfer, the Fe sublattice acting as a reservoir. The absence in ref. [3] of interaction between the sublattices inhibited the possibility of the disorder in the A sublattice to modify the local 3d density of states. In the present work we suggest an intermediate description staying between the first-principles approach and the previous oversimplified Correspondence to: Dr. N.A. de Ohveira, Centro Brasileiro de
Pesquisas Ffsicas (CBPF-CNPq), Rua Dr. Xavier Sigaud 150, Urea, Rio de Janeiro, RJ 22290, Brazil. ’ IF-UFRGS/CBPF-CNPq. 0304-8853/92/$05.00
model [3]. Electronic structure models usually are based on a tight binding (TB) description of the electronic states, recently revived by the use of the LMTO formalism [4]. This is a parameter free theory, but as mentioned before, the case of pseudobinqries with several atoms per cell and including disorder requires a larger computational effort and some technical difficulties to be overcome. We thus decided to adopt a simple two-sublattice model [51 and a convenient parametrization scheme to deal with the following points determining the magnetic properties: (a) Substitutional disorder in one of the sublattices, as described within the coherent potential approximation (CPA), and its role in modifying the charge transfer between the sublattices. (b) The relation between the ferrimagnetism of these materials and the interactions between sublattices. In the following we present the model, the parametrization procedure, and an application to the magnetic properties of (Zr,_,Hf,)Fe, pseudobinaries. Emphasis is given on the computational simplicity of the model and its capability of accounting for the magnetic data. The model may be tested using the formalism [6] proposed by Heine et al. for magnetostriction phenomena, when experimental data will be available. Also, this calculation is used as the starting point for
0 1992 - Elsevier Science Publishers B.V. All rights reserved
N.A. de Oliveira, A.A. Gomes / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
284
the finite-temperature description of these compounds via the functional integral formalism [7].
2. Formulation The pseudobinary compound (Zr 1_-xHf x)Fe, contains two interpenetrating sublattices, and Hf impurities are selectively introduced only in the A sublattice of the AB, compound. We describe this system using the TB formalism within the simplifying approximation of five identical local d-subbands. Let us define the corresponding Hamiltonian:
ULni f ni 1 + C $%,~aj,
+ C jj’a
TJFa,~ajvu + C U~nj t nj,
+ C
~~(UifolZic,
j U~Ujm),
eikRii.
(2c)
(1)
ijo
i = Zr or Hf,
E; = c; + U&r_,),
where a,‘, (aj,) is the creation (destruction) operator of one electron at site “9 (“j”) with spin u in the A (B) sublattice. The energy of a d-level at site i of the A sublattice, depending on the occupancy of this site, can take values E? or co”‘. In the B sublattice it can take only the value l r, since impurities are selectively dissolved in these materials. The terms Tiy (Tjy) represent hoppings between “ii”’ (“jj”‘) within the same A (B) sublattice, TF are hoppings between the A and B sublattices. Finally, the Coulomb interaction strengths are introduced in the Hamiltonian through the parameters Up, UAHfand Up. Some approximations are introduced now. (a) Firstly, since Hf and Zr are 5d and 4d elements, respectively, one has different local bandwidths, and thus off-diagonal disorder. To include in the simplest way off-diagonal disorder, an x-dependent bandwidth, given by the virtual crystal approximation may be adopted: A/,(X) = (1 -x>
q;:4” = CT,
between the corresponding sites and the k-dependent functions E:, E; and rk will be defined below. cc> The Coulomb interaction is dealed with using the Hartree-Fock approximation. This reduces the problem to define effective spin dependent energy levels of Hf, Zr and Fe through:
iv
+
P)
Riir, Rjjr and Rij are the separation
ii’a
+ C i
(2a)
k
H = ~&zi+,aim + c Ti~ai+,ait, ia
comparable bandwidths, as will be estimated below. The local bands associated with Zr and Hf atoms located at the A sublattice and centered at energies EF and crf, are then taken to have a common width A,(x). (b) The second approximation concerns a model description of the two-sublattice system, defined by [5]:
eFe 0” + UFe(nj_u). I7 = l
(3b)
The substitutional disorder introduced in the A sublattice is described within the CPA approximation. The levels EL in eq. (3a) are replaced by an effective medium J$,(z>, with z = E + ia, 6 -+ O+. This effective medium is determined self-consistently as follows: within the CPA description the Hamiltonian for an A sublattice configuration of effective atoms, except a given Zr or Hf atom occupying the origin, taken in the A sublattice is: H=H,+V*,
(4a)
where ia + C
ii ‘17
e,Fea,Lajo + C Tj~al~aj~q
(4b)
A,, +x Am,
where the A’s are Zr and Hf local bandwidths. This is reasonable since both Zr and Hf have
(3a)
VA=
(4c)
N.A. de Oliueira,AA. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x HfJFe,
The Green function (or resolvent) for the system corresponding to the Hamiltonian (4) is defined by a (2 x 2) matrix, reflecting the two-sublattice character of the material:
G=($I“,::)y G=g+gVG,
The resolvent becomes then: 2
-&(z) %-YEk
(64 from which one obtains performing braic operations:
w
Ly(1 -$/a)(E”,-E”) X[(Z-•:-E?)F(K)
( 1 VA 0 0
some alge-
1 F*a(Z)=gi~=
where the potential is given by the matrix: v=
--‘?I(
(5a)
and should satisfy the matrix Dyson equation:
285
0
-(z-e:-EO,)F(E”,)],
and g = (z - H,)- ’ is the unperturbed (2 X 2) matrix Green function for the Hamiltonian defined in terms of the effective medium. Using eq. (2) the unperturbed Green function in reciprocal space can be simply writen as:
(6d) 1 F,B(z)=g;;=
(Y(1 - y2/(Y)( E”,-E”_)
x[(z-i&,-aE’T)F(EI) -(z-Zo.-aE”,)F(E”,)], (he)
(6a) In order to simplify the numerical calculation of the unperturbed resolvent, let us introduce the following model for the two-sublattice system (2):
where 1 E”,= 2a(l -$/(Y) x
(1) E;=Q, (2) Et = ffEk) (3) r, =r,
[z-z,-2r,y+++)] t
+--z,+2r,y++-~,Fe)]2
(6b)
+ YEk,
where lk is a dispersion relation defined later on. In order to understand the physical meaning of the above approximations, let us recall the principles of an LMTO-TB calculation. Two different ingredients appear in a two-sublattice system: first, one must introduce structure functions associated, respectively, to the A and B sublattices; these structure functions reflect the geometric characteristics of the intermetallic. Second, three potential parameters [4] are associated with the involved atoms Fe, Zr and Hf. Thus the main advantage of this homothetic band model is to replace the structure functions by a single one and to incorporate in the parameters (Y, r, and y, the intra- and inter-sublattice hopping strength.
-4[(~-Z,)(~-~,Fe)-r~](~~2)}~"} (69 and F(z)
= /p(
l)/(
z-
l) de.
(6g)
The results (6), being analytical expressions for the local Green functions, simplify the numerical calculations, in particular for AB, systems with six atoms per unit cell. The general solution of Dyson’s equation (5b) in real space is:
v-a)
286
N.A. de Oliueira,A.A. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x HfJFe,
The local Green function for an atom i = Zr or Hf embedded in the effective medium follows from eq. (7a):
cY4
Gzw = 1 -
Using the standard procedure, is obtained from:
(‘-‘) 1-
(7b)
(EL -&(z))F,A(z)
the self-energy X,,
P-&(Z)
mre = 5( n: - .?J,
( 1Ob)
the factor 5 accounting for the degeneracy of the d-band. The magnetic moment per formula unit of the pseudobinary is given by: +xm,,
+ 2mFe.
( 1Oc)
3. Results of the numerical procedure Since the pseudobinary (Zr,_,Hf,)Fe, involves substitution of isoelectronic elements in the A sublattice, the total number of electrons should be independent of the concentration. Consequently any change in the average occupation number of the A sublattice will imply in a charge transfer between the A and B sublattices. The total number of electrons for a Laves compound is given by:
EF
/
Piu(E)
nF= = EFP,Fe (e) (T /
de, de,
(9a)
n,=&(X)
(9b)
where the occupation number of the A sites is given as a function ot the concentration x: by:
where or is the Fermi level and the local densities of states are:
piv(e)
( 1Oa)
[&Jz)]F,A(z)
Formally this is a quite similar result to that corresponding to the usual description of simple alloys; the important difference is contained in eqs. (6) for the functions F/(z) and F:(z 1, where the coupling between sublattices is included. This extends the early description [3], because disorder now dktorts also the B sublattice density of states. The number of electrons is obtained from:
pp=
the
mi = 5( IZ~~- IZ~_~), i = Zr or Hf,
m = (1 -x)m,,
(8)
ni, =
precision window. After self-consistency, magnetic moments are calculated from:
GTrn = GIrn
F:(z), G,4p,“‘( z),
(SC) i=
Zr or Hf.
(94 Eqs. (8), (7b), (91, (6), and (3) must be solved self-consistently. The procedure by which these coupled equations are solved is the usual one: given an initial set of parameters (Y, y, r,, U?, UAHf,and UFe, the dispersion relation Ed and values for the occupation numbers nk, using these equations one obtains an initial value for the self-energy 2,. One then recalculates the corresponding occupation numbers. This process is repeated until self-consistency, within a 0.002
n:(x)
+2&(x),
= (1 -.x)+,(x)
( lla)
+xn,,(x),
(lib)
and the occupation numbers n,(x), i = Zr, Hf or Fe are given by eqs. (9). We decided to start our fitting procedure from the x = 0 limit of the series because: (i) The density of states P(E) for the Fe sublattice to be used in eq. (6g) is obtained from the ZrFe, calculation of Terao and Shimizu [8], for the Cl5 structure, keeping only the main features and normalizing to unity (see fig. 1 for a comparison). (ii) The occupation number at the Fe site has been also extracted from the ZrFe, calculation
Bl. (iii) Finally and more importantly, it is well known that HfFe, shows coexistence of the Cl4 and Cl5 phases, which complicates the procedure of the fit.
N.A. de Oliveira,A.A. Gomes / Low-temperaturemagneticpropertiesof (Zr, _ x Hf,)Fe,
Fig. 1. Local 3d density of states as calculated in ref. [S] (dashed line) and simplified one (full line).
Since we use the homothetic model, eq. (6b), in performing the numerical calculations we need an alternative estimate for n.,(O) = n$‘i and in the opposite concentration limit, n,,(l) = n$& It should be remembered that compounds of Hf and Zr are isoelectronic, and this should be ensured during the calculation for any value of x. To estimate the number of d-electrons per site for the specific sites occupied by Zr or Hf atoms in the extreme concentration limits, we adopt the following procedure, as dictated by recent literature [2]. We approximate the local bands by constant and normalized densities of states, their centers and widths being estimated from the parameters of the self-consistent LMTO-ASA calculations tabulated in ref. [4] for the pure elements. These are given by:
lre = 0 7
A,, = 2,
lZr = 1.39,
A,, = 4.4,
EHf = 1.77,
A,, = 5.1.
Converting these quantities to reduced units where the Fe half bandwidth is taken equal to 1, and assuming that the number of electrons at the Fe site in the pure Cl5 compound is given by ref. [81 ($‘L = 1.44 = (7.2/5)), one gets er = 0.44, in our units. This choice implies in the following electron occupation numbers at the Hf and Zr sites: n$$ = 0.48 and $‘i = 0.56. Note that the total and concentration independent number of
287
electrons nT is obtained from eq. (111, using these values for x = 0. Our estimate of these occupation numbers indicates that charge transfer should occur between the A and B sublattices in order to restore the isoelectronic character. This transfer is automatically included in the formalism through eqs. (11) and (9); our estimate of the occupation numbers is only needed in order to fix, in the paramagnetic phase, the values of E? and ~0”’ for the compound, given the Coulomb interaction strengths. Within the description in terms of dominating diagonal disorder, the bandwidth of the local bands of Zr and Hf are taken equal. Thus the dispersion relation for the A sublattice in the homothetic model, eq. (6b), is proportional to that of the Fe sublattice, with average scaling factor (Y= 2.2. If a more rigorous approach is needed, a scaling factor which is concentration dependent due to the virtual cristal approximation, should be included. The Coulomb interaction parameters are chosen in order to reproduce the experimental magnetic moments of the pure intermetallic ZrFe,. The results for these parameters in our units are: Ur’ = 1.5 and Uz’ = 1.0; for Hf we have assumed UHf = 0.7. These values are consistent with the increase in extent of the d-wave functions in passing from Fe to Hf. Using these values of the Coulomb interaction parameters and the occupation numbers, the energy levels ~5 in eq. (3) may be fixed for decoupled lattices. We obtain E? = 1.963 and l f’= 2.198 and the so obtained local densities of states are shown in figs. 2 and 3. Now we turn on the inter-sublattice coupling, approximated in eq. (6b) by a constant hybridization value r, (see ref. [53> plus a k-dependent term. The constant term r,, corresponds to the usually adopted hybridization in model Hamiltonians. The range of inter-sublattice hopping in eqs. (2~) and (6b) is controlled by the k-dependence of lk; the parameter y in eq. (6b) defines the relative importance of the “hopping” between the A and B sublattices and the constant hybridization term. It has been shown by Yamada and Shimizu [9] that the hybridization between the spin polarized
288
N.A. de Oliueira, A.A. Games / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
150\
DENSITY
OF STATES
L--_ r3.0-
- - -4%
> 5.0 ENERGY
1.50 f
Fig. 2. Local densities of states for ZrFe* in the paramagnetic phase. Local 3d density of states (full line), local 4d (dashed line). The position of the Fermi level is indicated by the vertical line.
3d-states and the higher-energy states associated with the 4d or 5d atoms plays the central role in explaining the occurrence of ferrimagnetism in these compounds. Since hybridization in their paper may be interpreted as transferring electrons mostly confined to one of the sublattices to the other, we hope to reproduce the results of ref. [9] by suitably parametrizing the r, and y values. In ref. [9] the splitting of the up and down 3d-states implies that admixture, at the A sublattice sites, of the down spin states is larger than for up spin ones. Thus, the A sublattice down spin occupation number is expected to be larger than for the up spin case. These remarks suggest to allow for spin dependent values of the parameters r, and y, which we refer to as r,,r, roL and y f, y 1. We considered two extreme limits in order to obtain ferrimagnetic solutions r,‘/r,,l f 0 and y ?/y L = 0 or r,,T/rO1 = 0 and y T/y 1 # 0. These are the central parameters to establish ferrimagnetism through the unbalance of the interactions of different spins. The used values in this calculation are the following: r,f = 0.1, r,l = 0.9 or y T = 0.7 and y 1 = 0.2.
These particular choices of the parameter ratio can be roughly understood as follows. The k-independent parameter mixes states associated with the A and B sublattices, independently or real space considerations. If one intends to increase the occupation number of a given spin state, the more states are admixed, the larger the occupation number. On the contrary, according to eqs. (2~) and (6b), for a given k-dependence of lk, the larger the value of the parameter y, the larger the probability of a given spin electron to hop out, off the original A sublattice. The reverse hopping, namely from B to A, is in a way combined with the effect of a larger 3d Coulomb interaction in the B sublattice, which tends to conserve the difference in spin occupation numbers at B, which could be altered by the inter-sublattice hopping. To conclude, there is an effective inter-sublattice hopping which is larger from A to B and prefers up spin electrons. Thus at the end of the process there is smaller occupation of up than down spins at the A sublattice, thus ferrimagnetic behavior. In order to numerically verify how the relative strengths of the two described processes affect the existence of ferrimagnetic properties of the
DENSITY
OF STATES
Fig. 3. Local densities of states for HfFe, in the paramagnetic phase. Local 3d density of states (full line), local 5d (dashed line). The position of the Fermi level is indicated by the vertical line.
289
N.A. de Oliveira, A.A. Comes / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
Table 1 Magnetic moments as a function of the coupling strengths
compound, we have performed calculations using a sort of linearly interpolated parameters defined as:
P
r&
ror
YT
Y1
mz,
mFe
or mHf
Zr compounds 0.10 0.01 0.09 0.1 0.08 0.2 0.07 0.3 0.06 0.4 0.05 0.5 0.04 0.6 0.03 0.7 0.02 0.8 0.01 0.9 0.00 1.0
where F{(p) = (1 - p)T,” and y”(p) = py”. The two opposite limits are thus recovered, and intermediary values are thus obtained, which provides the crossover between the processes. The results of this calculation are shown in table 1. In this table one sees how critical the strengths of the two processes are in passing from ferrimagnetism at the ends of the range of p to ferromagnetism for intermediate regions. The numerical results show that for smaller values of the coupling parameters ferrimagnetic solutions are unstable. To verify this statement we calculated two “pure” intermediate situations: (a) TOT= 0.05, raL = 0.45 or y t = 0.0 and y J = 0.0 and (b) rof = 0.0, r’,l = 0.0 or y T = 0.42 and y L = 0.12. Case (a) is weakly ferromagnetic, whereas case (b) is weakly ferrimagnetic. These are values separating stability regions. In fig. 4 the variation of the magnetic moments in the A sublattice is shown as a function of the concentration. Note that the Zr and Hf magnetic moments at the A sites are always antiparallel fo the Fe magnetic moment. This calculation was performed using pure hopping-like interactions.
-I 4.00
Hf compounds 0.10 0.0 0.09 0.1 0.08 0.2 0.07 0.3 0.06 0.4 0.05 0.5 0.04 0.6 0.03 0.7 0.02 0.8 0.01 0.9 1.0 0.00
0.90 0.81 0.72 0.63 0.54 0.45 0.36 0.27 0.18 0.09 0.00
0.00 0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.70
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
+ 0.49 - 0.28 - 0.08 0.08 0.20 0.24 0.19 0.03 -0.10 - 0.34 - 0.50
1.67 1.75 1.87 1.98 2.10 2.18 2.20 2.10 2.03 1.98 1.90
0.90
0.00
0.00
0.81 0.72 0.63 0.54 0.45 0.36 0.27 0.18 0.09 0.00
0.07 0.14 0.21 0.28 0.35 0.42 0.49 0.56 0.63 0.70
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
- 0.52 - 0.33 -0.16 - 0.03 0.06 0.14 0.07 -0.00 -0.16 -0.30 - 0.41
1.50 1.57 1.65 1.73 1.94 1.94 1.97 1.99 1.87 1.76 1.69
In fig. 4 the average magnetic moment of the sublattices and the total magnetization of the
f ’ + ’ + e,
-m 5g 3.00 2.00 I
q
0
0
Q
0
cl
$ 0 t;
z
2
1.00
t
0.00 .
0
0..
I
I
I
I
0.2
0.4
06
0.8
.o
l
0
0
0
I
l
1.0 CONCENTRATKH l mn
X1
-1.ooL Fig. 4. Magnetic moment of the pseudobinary compound (in Bohr magnetons kb): mA and 2mFe are the magnetizations at the A and B sublattices, respectively.
N.A. de Oliveira, A.A. Games / Low-temperature magnetic properties of (Zr, _ x Hf,)Fe,
290
Table 2 Magnetic moments as a function of concentration X
“2
0.0
- 0.505 -0.510 -0.510 - 0.520 - 0.520 - 0.535 - 0.550 - 0.550 - 0.555 - 0.560 - 0.570
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mHf
- 0.383 - 0.385 - 0.385 - 0.390 - 0.390 - 0.395 - 0.400 - 0.400 - 0.405 -0.410 - 0.415
mA
-
0.505 0.497 0.485 0.481 0.468 0.465 0.460 0.445 0.434 0.425 0.415
parameters as used in the construction are used here.
mFe
mAB2
1.900 1.875 1.840 1.830 1.805 1.790 1.750 1.735 1.725 1.710 1.690
3.295 3.253 3.195 3.179 3.142 3.115 3.040 3.025 3.016 2.995 2.965
pseudobinary are shown, the results being indicated in table 2. In fig. 5 the spin polarized density of states is shown for the concentration x = 0.5. This figure illustrates the effect of the spin-dependent hopping in producing an antiparallel magnetic moment at the A sites. The same
of fig. 4
4. Conclusions Using a parameterized calculation it was possible to account for the magnetic properties of (Zr, _,Hf,)Fe, compounds. The advantage of this parametrization was that it clearly exhibits the relevance of each parameter for the concentration dependence of magnetic moments and more importantly for the occurrence of ferrimagnetim; thus this may serve as a guide for more precise and detailed ab initio calculations. Also, such a calculation enables the functional integral description of the temperature dependence of the magnetizations, which becomes simpler if the one-electron densities of states are extracted from this parametrized version.
Acknowledgements 1.50
DENSITY
OF STATES
The authors would like to thank C. da Silva for discussions and a critical reading of the manuscript.
References
1.501 ” Fig. 5. Spin-polarized local densities of states for the concentrations x = 0.5. The position of the Fermi level is indicated by the vertical line.
[l] F. Baudelet et al., Europhys. Lett. 13 (1990) 751. M. Iijima et al., J. Phys.: Condens. Matter 2 (1990) 10069. [2] 0. Eriksson et al., Phys. Rev. B 40 (1989) 9519. [3] L. Amaral, F. Livi and A.A. Comes, J. Phys. F 12 (1982) 2213. [4] O.K. Andersen et al., in: Highlights in Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (NorthHolland, Amsterdam, 1985). [5] J. Giner et al., J. Phys. F 6 (1976) 1281. [6] V. Heine et al., J. Magn. Magn. Mater. 43 (1984) 61. [7] N.A. de Oliveira and A.A. Comes, to be published. [8] K. Terao and M. Shimizu, Phys. Stat. Sol. (b) 139 (1987)