Non-uniform RKKY magnetism in thin wires

Physics Letters A 372 (2008) 5484–5487

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Physics Letters A www.elsevier.com/locate/pla

Non-uniform RKKY magnetism in thin wires E.Z. Meilikhov ∗ , R.M. Farzetdinova Kurchatov Institute, 123182 Moscow, Russia

a r t i c l e

i n f o

Article history: Received 15 June 2008 Accepted 18 June 2008 Available online 20 June 2008 Communicated by V.M. Agranovich

a b s t r a c t Analytical mean-field continual model of RKKY (Ruderman–Kittel–Kasuya–Yosida) magnetism in thin (nanosized) wires is considered. The spatial distribution of the magnetization is described by the nonlinear integral equation. Its solution delivers the radial magnetization distribution that occurs to be not only highly non-uniform but significantly non-monotone as well. The concrete form of the distribution depends on the wire radius, the interaction length, and concentrations of magnetic impurities and itinerant charge carriers. Results could be used for describing properties of nanosized systems based on the diluted magnetic semiconductors. © 2008 Elsevier B.V. All rights reserved.

Indirect magnetic impurities’ interaction of Ruderman–Kittel– Kasuya–Yosida (RKKY) type is considered as one of the basic mechanisms of the magnetic ordering in systems with free carriers of high concentration (nonmagnetic metals or degenerate semiconductors with magnetic impurities). Because every potentially interesting electronic device is characterized by nanosizes it is actual to consider how magnetic features of the relevant systems depend on their finite size [1–3]. The typical instances are thin wires of the diameter on the order of tens lattice constants that are just considered in the Letter. Under considering the processes of ordering interacting magnetic moments in small systems one should take account of some new (as compared to the “large” systems) circumstances associated with the surface and quantum-size effects. Firstly, with indirect interaction (of RKKY type) putting into effect with the aid of free charge carriers it is necessary to take into account the modification of the carrier energy spectrum connected with the system finiteness. Secondly, the non-uniformity of the effective field describing the interaction of spins comes to be essential as those fields in the bulk and near the surface differ. In the system of the finite size, there are three parameters of the length dimensionality: the system size R by itself, the carrier de-Broglie wavelength λ F = 2π /k F defined by their concentration, and the characteristic interaction length l, coinciding, in the simplest case, with the carrier free path. Depending on the relation between those parameters, the following effects are possible and essential: (i) increasing the surface contribution—becomes important at R  , (ii) the Fermi energy rise associated with the finite system size—appreciable at R  λ F , (iii) shifting the lower boundary of the carrier wave num-

*

Corresponding author. E-mail address: [email protected] (E.Z. Meilikhov).

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ber interval due to the level quantization—comes to be essential at R 2  λ F . The last described factor “works” in systems with the relatively large carrier free path, and its effect on the RKKY interaction features has been studied in [1]. It is essential if   k F R 2 only. Below we have in mind systems like diluted magnetic semiconductors where carrier mobilities (and, hence, carrier free paths) are small. It is easily checked that with the mobility on the order of 10 cm2 /V s (which is typical for the diluted GaAs(Mn) thin films [4]) the cited inequality is not fulfilled if R  10 Å. This means that the wire should not be considered as the one-dimensional object with the inherent quantization of energy levels. In the case considered they are drastically broadened by collisions, and the system grows effectively three-dimensional. Therefore, we consider first two listed effects only. To obtain concrete results we use the mean-field theory extended over the case of the non-uniform effective magnetic field. Early, similar approach has been exploited for thin magnetic films with the model spin interaction of the Ising type [5,6] or the films with surface magnetic anisotropy [7–9]. Our approach is developed for considering systems with the distance dependent spin interaction. Specifically, we study the interaction of the RKKY-type, characteristic of the diluted magnetic semiconductors. It is known that for RKKY interaction, the magnetic state with all spins being parallel (antiparallel) to each other, could not be, in general, the ground state [10]. However, wires considered in the Letter are believed to be strongly magnetically anisotropic. Accordingly, we consider two variants only: (i) easy axis, and, hence, the local magnetization are everywhere parallel to the wire axis, and (ii) “vortex” structure with the azimuthal local magnetization. Corresponding to the infinite system, the standard result [10] for the energy J (r ) of RKKY interaction between spins S1 , S2 spaced at the distance r reads W (r ) = J (r )S1 S2 with

E.Z. Meilikhov, R.M. Farzetdinova / Physics Letters A 372 (2008) 5484–5487

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J (r ) = − I 0 Φ(r )e −r /l ,

 4 a

Φ(r ) =





ϕ (r ) cos ϕ (r ) − sin ϕ (r ) ,

r

(1)

2

where I 0 = ( ma2 J 2pd )/32π 3 , J pd is the energy of the “contact” inh¯

teraction of atomic spins and itinerant electrons (holes), ϕ (r ) = 2k F 0 r, a is the lattice constant, k F 0 = (3π 2 p )1/3 is the Fermi momentum of carriers of the concentration p for the infinite system. Note that small l suppresses effectively the oscillatory J (r ) behavior. For the finite system, one needs to take into account the dependence k F ( L ) of the Fermi wave number on the system size. There is no difficulty to get the relationship [1] k F (L) = k F 0 +

α L

,

α ∼ 1.

(2)

on a given The local effective RKKY-field H RKKY , generated ∞ spin S0 , is defined by the relation μ H RKKY = i =1 J (r i )S0 Si where r i is the distance from that spin to the ith magnetic impurity, μ is the impurity magnetic moment. The distance r i might not be less than the spacing rmin between two adjacent lattice sites allowable for the magnetic impurities. √ (For the diluted magnetic semiconductor Ga1−x Mnx As, rmin = a/ 2.) In the continual approximation, which applicable if the impurity distribution is uniform, the sum could be replaced by the integral



μ H RKKY (r) = nμ

J (r − r ) j (r ) d3 r ,

(3)

where the integration is spread over the volume occupied by impurities, nμ is their concentration, and 0 < j < 1 is the reduced magnetization in the point r . Contrary to the infinite system, the value of that integral depends on the specific geometry of the system and the position of the considered point. Let us, firstly, consider the thin cylindric wire of the radius R, magnetized along its axis z. In that case, it is convenient to write the expression (3) for the effective magnetic field in the point being at the distance h from the wire axis, in the cylindrical coordinates:



μ H RKKY (h) = nμ

J (ρ ) j (r )r dr dz dφ,

(4)

ρ rmin

where ρ = (r 2 + h2 + z2 − 2rh cos φ)1/2 is the distance from a given point to the small element of the wire with coordinates r , z, φ and magnetization j (r ). The integration is executed over the whole wire volume taking into account that the distance between magnetic impurities could not be less than rmin . After replacing the integration variable z by the variable ρ , it follows from Eq. (4)

R

μ H RKKY (h) = I 0

K (r , h) j (r ) dr ,

(5)

r =0

where K (r , h) =

2nμ r I0

2π ∞ ρmin

0

ρ J (ρ ) ρ 2 − r 2 − h2 + 2rh cos φ

dρ dφ,

(6)

ρmin = ρmin (r , h, φ) = Max[(r 2 + h2 − 2rh cos φ)1/2 , rmin ]. Then the standard mean-field equation

 j (h) = tanh

μ H RKKY (h)



kT

transforms into the integral equation

(7)

Fig. 1. Radial distributions of the axial magnetization in the wire of the radius R = 10a at various temperatures τ (the inset). Accepted parameters’ values:  = 3a, k F a = 1, 4π nμ a3 = 1. The main panel: the temperature dependence of the average magnetization  j  for that wire.

j (h) = tanh

1

R

 K (r , h) j (r ) dr ,

τ

(8)

0

where τ = kT / I 0 is the reduced temperature. The relation (5) defining the local field H RKKY (h) in any point of the wire is non-local: that field is the functional of the local magnetization j (r ) and is defined by all wire points. The nonlinear integral equation (8) determines the spatial distribution of the magnetization in the system for a given temperature. To solve it we have used the method of successive approximations R putting j i (h) = tanh[(1/τ ) 0 K (r , h) j i −1 ( z) dr ], where j i (h) is the ith approximation (i = 0, 1, 2, . . .). The starting distribution has been put as uniform one: j 0 (h) = Const. Calculations show that magnetization distribution is highly non-uniform: only the paraxial section of the wire has a significant magnetization, while its periphery is practically non-magnetic (cf. the inset in Fig. 1). In addition, Fig. 2 shows that the magnetization of wires which radius is comparable with the interaction length (/ R ∼ 1) is not only highly non-uniform but significantly nonmonotonous as well. For the infinite interaction length ( → ∞), the distribution of the magnetization in the wire is practically uniform. One could characterize the non-uniformly magnetized wire by R the average magnetization  j  = (1/ R ) 0 j (r ) dr. Temperature dependence of that quantity for the wire of the diameter 2R = 10a is also represented in Fig. 1. That allows to determine the Curie temperature τC , defined as the temperature for which  j  → 0. In the example considered it equals τC ≈ 0.95. The distinguishing feature of the dependence corresponding to the finite  is its negative curvature. At  → ∞, the temperature dependence of the magnetization is of the standard appearance with the positive curvature. Near the Curie temperature, j → 0 and Eq. (8) is simplified, conversing to the homogeneous linear integral equation: j (h) = R (1/τC ) 0 K (r , h) j (r ) dr, wherefrom it follows that the temperature τC is nothing but the eigenvalue of that equation kernel. One could easily expand the consideration for the case with the non-zero external magnetic field H ext directed parallel to the wire axis. To do that, it is sufficient to introduce the additional term

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E.Z. Meilikhov, R.M. Farzetdinova / Physics Letters A 372 (2008) 5484–5487

Fig. 2. Radial distributions of the axial magnetization in the wire of the radius R = 10a at various interaction lengths . Accepted parameters’ values: τ = 0.2, k F a = 1, 4π nμ a3 = 1.

Fig. 4. Radial distributions of the azimuthal magnetization in the wire of the radius R = 10a at various temperatures τ (the inset). Accepted parameters’ values:  = 3a, k F a = 1, 4π nμ a3 = 1. The main panel: the temperature dependence of the toroidal magnetic moment T for that wire.

for the spherical particle the role of the surface is even higher. However, as we have shown early [1], even in that case the quantum modification of the RKKY interaction due to the finite system size becomes to be important for very small particles only (R  5a, see Fig. 2 in [1]). That is why for wires with R  10a one could neglect above mentioned quantum corrections (really, we have taken into account Eq. (2) only). Now, turn to the wire with the azimuthal magnetization. Formally, this case differs from the above in that the additional factor cos φ appears in the expression (5) for the effective magnetic field. Then the kernel of the integral equation (8) reads

K (r , h) =

2nμ r I0

2π ∞ 0

ρmin

ρ J (ρ )

ρ 2 − r 2 − h2 + 2rh cos φ

dρ cos φ dφ, (9)

Fig. 3. Radial distributions of the axial magnetization in the wire of the radius R = 10a at various magnetic fields. Accepted parameters’ values: τ = 0.8, k F a = 1, 4π nμ a3 = 1,  = 3a.

μ H ext /τ I 0 in the argument of the tanh function in Eq. (8). The evolution of the spatial distribution of the magnetization under the action of the external field and the field dependence of the average magnetization could be found. The relevant results are presented in Fig. 3. As could be seen, the magnetic state characteristics of thin wires differ significantly from those of the bulk. The specific reasons are (i) the comparability of the interaction length and the wire radius, and (ii) the high surface-to-volume ratio. Notice, that

wherefrom it immediately follows that K (r , 0) = 0 and, hence, j (0) = 0 (cf. (8)). Thus, the magnetization at the wire axis equals zero in that case. Besides, it is depressed near the surface, too. That means the magnetization is maximum somewhere in the wire bulk. Though the highly non-uniform radial distribution of the magnetization might be, as previously, characterized by the average magnetization  j , one should see that now the total magnetic moment of the wire equals zero and the proper magnetic parameter is the toroidal magnetic moment T ∝ [r × j(r)] dV [11]. In the inset of Fig. 4, radial magnetization distributions for the wire with the azimuthal magnetization are represented. Those distributions are highly non-uniform even in the medial section of the wire. As for the peripheral and the paraxial areas, they are practically non-magnetic. In the same Fig. 4, the temperature dependence of the toroidal magnetic moment T is shown. Analogously to the temperature dependence of the average axial wire magnetization  j (τ ) (see Fig. 1), T(τ ) dependence has the negative curvature. With increasing the interaction length , the non-uniformity of the azimuthal magnetization distribution decreases (in a manner like it is shown in Fig. 2), so for the infinite interaction length ( → ∞) that distribution is almost uniform.

E.Z. Meilikhov, R.M. Farzetdinova / Physics Letters A 372 (2008) 5484–5487

In conclusion, we have considered the mean-field continual model of the RKKY magnetism of thin wires under the condition of their essential non-uniformity. We succeeded in discovering the spatial magnetization distribution defined by the nonlinear integral equation using some iterative procedure of rapid convergence. At temperatures close to the Curie temperature, that integral equation comes to be linear, the Curie temperature by itself is determined as the eigenvalue of the equation kernel. Both, for wires with the axial and azimuthal magnetization its profile is significantly non-uniform if the interaction length is comparable with the wire radius. Temperature dependencies of the average wire magnetization or the toroidal magnetic moment have been obtained. Results could be used for describing properties of nanosized systems based on the diluted magnetic semiconductors. Acknowledgement This work has been supported by Grant # 06-02-116313 of the Russian Foundation of Basic Researches.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

E.Z. Meilikhov, R.M. Farzetdinova, Phys. Rev. B 75 (2007) 052402. T. Balcerzak, J. Mag. Mag. Mater. 310 (2007) 1651. N.A. Usov, S.A. Gudoshnikov, J. Mag. Mag. Mater. 290–291 (2005) 727. D. Neumaier, M. Schlapps, U. Wurstbauer, J. Sadowski, M. Reinwald, W. Wegscheider, D. Weiss, arXiv: 0711.3278. R. Pandit, M. Wortis, Phys. Rev. B 25 (1982) 3226. D.I. Uzunov, M. Suzuki, Phys. A: Stat. Theor. Phys. 204 (1994) 702. D.L. Mills, Phys. Rev. B 39 (1989) 12306. R.C. O’Handley, J.P. Woods, Phys. Rev. B 42 (1990) 6568. A. Aharoni, Phys. Rev. B 47 (1993) 8296. D.C. Matiss, The Theory of Magnetism, Harper & Row, New York, Evanston, London, 1965. V.M. Dubovik, V.V. Tugushev, Phys. Rep. 187 (1990) 145.