Ferromagnetism induced by magnetic vacancies as a size effect in thin films of nonmagnetic oxides

Thin Solid Films 534 (2013) 685–692

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Ferromagnetism induced by magnetic vacancies as a size effect in thin films of nonmagnetic oxides Maya D. Glinchuk, Eugene A. Eliseev, Victoria V. Khist, Anna N. Morozovska ⁎ Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, 3, Krzhizhanovskyi Str., Kiev 03142, Ukraine Institute of Physics, National Academy of Sciences of Ukraine, 46, Nauka Ave., Kiev 03680, Ukraine

a r t i c l e

i n f o

Article history: Received 11 February 2012 Received in revised form 23 February 2013 Accepted 25 February 2013 Available online 13 March 2013 Keywords: Misfit strain Nonmagnetic oxides Magnetic defects

a b s t r a c t A theory of magnetization induction in thin films of binary nonmagnetic oxides has been developed. We show that the origin and the main peculiarities of magnetization experimentally observed in thin films of such nonmagnetic oxides as SnO2, CeO2, Al2O3, ZnO, MgO, and HfO2 can be explained if oxygen vacancies are considered as magnetization sources. Our calculations have shown that the oxygen vacancies become magnetic at the film–substrate interface, and a long-range ferromagnetic order appears in thin films at room and higher temperatures. The role of substrate turns out to be extremely important for the accumulation of magnetic vacancies taking place owing to the film-substrate misfit stress. The vacancy accumulation has been shown to result in the film magnetization increase, which agrees with experimental data. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Researches of magnetization in thin films of oxides that are nonmagnetic in the bulk were started in 2004, when Venkatesan et al. [1] reported about an unexpected observation of magnetism in thin HfO2 films on sapphire or silicon substrates. Later, the observation of magnetism in thin films of HfO2 and other oxides on various substrates at room temperature – the so-called d 0-magnetism [2–7] – attracted much attention of scientific community. The most investigated system seems to be thin HfO2 films on various substrates (see Refs. [2–4]). In Ref. [2], the films were produced by pulsed-laser deposition onto various sapphire cuts. Textures with different parameters were observed not only on different substrates, but even on the same substrate when the films were deposited under different conditions. In particular, a transmission electron microscopy of a film cross-section on the sapphire substrate revealed a columnar texture with a column diameter of about 8 nm. The image of the lattice in the interface region showed the availability of a fine-grained (~ 10 nm) polycrystalline texture. The magnetic moments of the specimens oriented perpendicularly or in parallel to the applied magnetic field were measured with the help of a superconducting quantum magnetometer, and the parallel magnetic moment turned out to be 19% lower at that. The extrapolated Curie temperature was much higher than 400 K; and the magnitude of magnetic moment, as well as the parameters of hysteresis loop, was shown to be dependent substantially on the substrate type.

⁎ Corresponding author. Tel.: +380 445252700. E-mail address: [email protected] (A.N. Morozovska). 0040-6090/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tsf.2013.02.135

The studies carried out in Refs. [3,4] allowed light to be shed on the origin of magnetic defects in thin nonmagnetic-oxide films, i.e. on the mechanisms of d 0-magnetism. Taking into account that all the HfO2, TiO2, and In2O3 films were colorless, shiny, and highly transparent, and the concentration of magnetic impurities in them was well below 10−2 wt.%, the large value of magnetization can hardly by attributed to any kind of impurities. That is why the oxygen vacancies at the interface between the film and the substrate were supposed to be the main source of magnetization (see, e.g., Ref. [4]). The disappearance of magnetization after annealing in oxygen atmosphere [7] confirms this supposition. A strong dependence of magnetic moment on the film thickness is another point to be emphasized. Really, the saturation magnetization for TiO2 and HfO2 films 10 nm in thickness was, respectively, 20 and 15 times larger than that for 200-nm films (see [3] and Refs. therein), i.e. this parameter was nearly reciprocal to the film thickness. Therefore, the number of magnetic defects (oxygen vacancies) in the film was independent of its thickness. This fact may be directly interpreted as evidence that the observed magnetization was induced by the defects mostly localized near the interface between the film and the substrate. Since the physical origins remain unexplained until now for the majority of the features described above, this work aimed at filling the gap (at least, partially) in our knowledge concerning the physics of d 0-magnetization in thin films, as well as at studying the appearance of ferromagnetic order at room and higher temperatures. We attempted to find why oxygen vacancies are mostly localized at the film–substrate interface, the magnetization depends on the film thickness, and the columnar structures emerges, as well as to elucidate the contribution of substrate into all the relevant phenomena.

M.D. Glinchuk et al. / Thin Solid Films 534 (2013) 685–692

The structure of the paper is as follows. The model and the corresponding basic equations are described in Section 2. In particular, we consider the reorientation of crystallographic axes in thin polycrystalline films under the action of film–substrate misfit strain (Subsection 2.1), the accumulation of vacancies near the film–substrate interface (Subsection 2.2), the magnetic state of vacancies at the film–substrate interface (Subsection 2.3), and the ferromagnetic long-range order in the films (Subsection 2.4). In Section 3, the relevant dimensionless variables are introduced and evaluated. The influence of size effect on the magnetic film properties is analyzed in Section 4. Discussion of the results obtained and their comparison with experimental data are presented in Section 5. 2. The model and the basic equations 2.1. Reorientation of crystallographic axes in polycrystalline thin films by misfit strain In this subsection, we will discuss the possibility for the crystallographic axes to be reoriented under the action of the film–substrate misfit strain. Actually, inhomogeneous elastic fields originating from the misfit strain um (driven by such factors as the lattice mismatch, the difference between thermal expansion coefficients, growth defects, and so forth) can induce the emergence of a texture in a polycrystalline thin film deposited onto a thick rigid substrate that is characterized by thermal properties and lattice constants different from those for the film. The in-plane strains u11 = u22 = um are inevitably present at the film–substrate interface and induce the out-of-plane strain u33 ¼ 2s12 um s11 þs12 either in the epitaxial film, where the strain does not relax [8,9], or near the boundary of dense polycrystalline thin film, where the strain relaxation should take place [10]. The anisotropy of out-of-plane and in-plain strains (u33 ≠ u11 ≠ 0) immediately leads to the appearance of tetragonal symmetry in the direction perpendicular to the film surface (below, this direction will be regarded as the c-axis) even in the case of unperturbed cubic lattice. The misfit-strain contribution to the tetragonality is evaluated as follows: c 1 þ u33 s −s −1 ¼ −1≈u33 −u11 ¼ 12 11 um : a 1 þ u11 s11 þ s12

ð1Þ

Besides the tetragonality, the anisotropic strains u33 ≠ u11 ≠ 0 can lead to the appearance of internal electric field Ezb (regarded as a built-in field), e.g., via the piezoelectric effect that inevitably presents in the vicinity of the surface of non-piezoelectric (in the bulk) mateS S rials [11]. The internal electric field is equal to Ezb ≅ d33 u33 + d31 u11 + S d32 u22 ~ um. Here, dijS are the coefficients of the surface piezoelectric effect tensor; they are nonzero near the surface owing to the inversion symmetry breaking in the z-direction. If we assume a low concentration or even the absence of screening carriers at the growth temperature, the field Ezb will penetrate into the film depth to make the orientation of crystallographic c-axis preferable in the direction perpendicular to the film surface. At temperatures much lower than the growth one, the c-oriented texture can serve as evidence of the internal electric field. Thus, we explained how the misfit strain can reorient the crystallographic axes and create textures in polycrystalline thin films. The film texture seems to play an important role in that the film magnetization is much higher than that in oxides nanoparticles [6,7,12,13]. Really, for a spherical nanoparticle on the surface, any orientation of its magnetic moment is possible with an equal probability, so that the average magnetization of the particles should equal zero. But in natural powders, the particle shape deviates from the spherical one. Moreover, none of technological methods allows nanoparticles to be fabricated with the same shape. As a result, powder includes up to a few percent of spherical particles [14]. Therefore, the magnetization in nanoparticle powders should expectedly be about some

hundred times lower than that in thin films. This conclusion is in agreement with experimental data [6,7,12,13]. 2.2. Vacancy accumulation near the film–substrate interface Consider an oxide film of thickness h made up of a nonmagnetic material. The film deposited on a rigid substrate with the cubic lattice. Let the film–substrate lattice-mismatch strain um appear at the boundary z = 0 (i.e. u11(z = 0) = u22(z = 0) = um), whereas the surface z = h is regarded to be mechanically free (i.e. the normal stress σ3i(z = h) = 0). Without the misfit strain relaxation, the elastic stress field σjk is spatially uniform inside the film [15] and looks m like [8] σ 11 ¼ σ 22 ¼ s11uþs and σ31 = σ32 = σ33 = 0, where sij are 12 elastic compliances. However, vacancies, as well as other topological defects, by accumulating near the film–substrate interface (see Fig. 1), lead to stress relaxation. The concentration of vacancies increases near the film surface, in particular, owing to a substantial lowering of their formation energy [16–18]. The corresponding density-functional calculations show that the energy of vacancy formation near the surface is lower than that in the bulk by about 3 eV for GaN [18] and by 0.28 eV for MgO [16]. Assuming that the stress field relaxes completely in the film depth due to the vacancy segregation, the inhomogeneous concentration of neutral vacancies, NV(r), can be obtained by minimizing the free energy FCS. The latter, besides the concentration-strain energy, also includes the contribution from the configuration entropy [19,20], so that
 3 V F CS ¼ ∫ d r βij ðNV ðrÞ−NV0 Þσ ij ðrÞ þ kB TNV ðrÞð lnðNV ðrÞ=NV0 Þ−1Þ : ð2Þ V

Here, NV0 is the vacancy concentration far from the interface, kB V the Boltzmann constant, T the absolute temperature, βjk the elastic dipole (in other words, the Vegard expansion) tensor for vacancies [21–23], and σjk(r) the stress redistributions appearing at the interface due to the misfit. The first term in Eq. (2), corresponds to the concentration-strain energy [19,20], and the second and third ones describe the contribution of configuration entropy [19,20]. Minimizing the free energy FCS according to the equation ∂ FCS/∂ NV = 0, we obtain

NV ðrÞ ¼ N V0 exp

! V βjk σ jk ðrÞ : kB T

εe

z

εi

ð3Þ

surface

Oxide film

z=h z=zmax z=hc z=0

εS

0

Ferromagnetic interlayer

686

Dielectric substrate

Fig. 1. Schematic diagram of an oxide film with magnetic vacancies (circles) accumulating at the film–substrate interface. The ferromagnetic interlayer thickness is about a few hc's.

M.D. Glinchuk et al. / Thin Solid Films 534 (2013) 685–692

Following Kim et al. [24], we assume that nonzero stresses vanish exponentially with the distance from the film–substrate interface, σ 11 ðzÞ ¼ σ 22 ðzÞ ¼

um expð−z=hc Þ ; s11 þ s12

ð4Þ

where hc ≈ 1 − 10 nm is the characteristic length of stress relaxation [24]. Since |um| ~ 0.01–0.05 and (s11 + s12) −1 ~ 10 12 J/m 3, we obtain |σii(0)| ~ 10 10 J/m 3 and |βijVσij(0)| ~ 4 × 10 −20 J. Under the condition h/hc > > 1, the relative concentration of neutral vacancies drastically changes at the film surface,  0
V 1   β11 þ βV22 um  NV ðz ¼ 0Þ h A ¼ exp@ 1− exp − NV ðz ¼ hÞ hc kB T ðs11 þ s12 Þ
 8 > βV11 þ βV22 um > > > 102 ; > > 5; < k
B T ðs11 þ s12 Þ ≈ V V > > β11 þ β22 um > > : <10−2 ; <−5: kB T ðs11 þ s12 Þ

ð5Þ

It is seen from Eqs. (4) and (5) that the influence of substrate on the accumulation neutral vacancies via the misfit strain um is extremely strong, because σii ~ um and  0
V 1   β11 þ βV22 um z A exp − NV ðzÞ ¼ N V0 exp@ hc kB T ðs11 þ s12 Þ

ð6Þ

(cf. Eq. (10) in work [25]). Thus, the higher are the strain um and the magnitudes of elastic dipole moments, the stronger is the vacancy accumulation/depletion at the film–substrate interface. Vacancies are accuV V mulated in the layer 0 ≤ z b hc if (β11 + β22 )um > 0. It is worth emphasizing that the parameter um depends on the film–substrate pair and the quantities βiiV characterize the vacancy type. In the general case, both the cation and anion vacancies in oxides can be the sources of magnetization [41,42]. We estimated the oxygen and cation vacancy concentrations in the framework of a simple model, in which the volume of corresponding ion can be taken for the elastic dipole magnitude [26]. The estimates give the following values: |βiiO| = 11.49 Å 3, |βiiHf| = 1.50 Å3, |βiiMg| = 1.56 Å3, |βiiIn| = 2.15 Å3, and |βiiTi| = 2.66 Å 3 [27]. Note that  1 0
O um 2 β11 −βcation 11 A: kB T ðs11 þ s12 Þ

O cation NV ð0Þ=NV ð0Þ≅ exp@

ð7Þ

Since |βiiO| > > |βiication|, one can see from Eqs. (3) and (4) that the concentration of oxygen vacancies at the interface has to be 2 to 3 orders of magnitude higher than the cation vacancy concentration for absolute values of um larger than 0.1%, an appropriate film–substrate V V pair with (β11 + β22 )um > 0, and at room temperature. Therefore, only the oxygen vacancies will be considered as a source of magnetization in the films. 2.3. Magnetic state of vacancies at the film–substrate interface and their direct exchange The magnetic properties of vacancies and a possible formation of magnetic phase in massive samples were examined in a number of studies [28–33]. The first-principles calculations [16,34–40] showed that the considered influence of ions in the nearest vicinity of cation vacancies can be the main reason for the vacancy-induced magnetism to emerge in binary oxides. The first-principles calculations also predict the magnetism induced by Ti and oxygen vacancies in the bulk of non-magnetic perovskite SrTiO3 [41]. Quantum-mechanical calculations [42] show that the ground states of such impurities as He, Li +,

687

Be 2+, and so on, as well as cation and anion vacancies, in binary solids are triplet (the spin Σ = 1) both at the surface and in its vicinity. Below, we focus our attention on the ferromagnetic exchange between magnetic neutral oxygen vacancies as a probable origin of ferromagnetic state near the film surface. A typical fraction of neutral vacancies is about 97% of their total number [43]. Our calculations are based on the analytical results obtained in works [42,44,45] in the framework of direct variational method. In particular, the method was used to solve the Schrödinger–Wannier equation in the effective mass approximation and making allowance for both the Coulomb interaction between the defects and charge carriers and the contribution made by image charges [43,46,47] near the film surfaces z = 0 and z = h. Since neutral defects are almost immovable, we introduce the following static dielectric constants: εi inside the film (0 ≤ z ≤ h), εe = 1 outside the film (z > h), and εS > 1 in the dielectric substrate (z b 0). The wave functions φnlm of the carriers localized near the surface defect in the solid material has to be almost zero in the dielectric environment (air or the dielectric substrate) owing to a high enough contact barrier, i.e. φnlm(x, y, z = 0) = 0 and φnlm(x, y, z = h) = 0. The coordinate dependences of localized electron wave functions were chosen as linear combinations of eigen functions φnlm(r) ~ z(h − z)Rnl(α|r − r0|) Ylm(θ,φ) of a hydrogen-like atom localized at the defect site r0 = (0,0, z0). Here, 0 ≤ z0 ≤ h, Rnl(r) are the radial functions, Ylm(θ,φ) are the spherical harmonics, and z = r cos(θ). The variational parameters, e.g., α, can be determined by minimizing the energy in the first order of conventional perturbation theory, with the electron–electron Coulomb interaction and all interactions with image charges being considered as a perturbation. The lowest wave functions of electrons localized at the neutral vacancies at the interface are constructed from the pz states, which transform into spherically symmetric 1s and 2s functions sufficiently far from the surface. The results of calculations of energy levels executed in the framework of conventional perturbation theory showed that the magnetic triplet state 2pz3pz appeared to be the ground one in the vicinity of the film surfaces, while the nonmagnetic singlet 2pz2pz is the ground state far from the surface (at zmax b z b h) [42]. The energy difference between the lowest triplet and singlet states depends on the distance z0, but is typically smaller than kBT at room temperature, T ≈ 300 K. According to Hund's rule, which orients the spins of two fermions in the same z-direction, the triplet state becomes magnetic. The ferromagnetic long-range order, when the spontaneous magnetization Mz differs from zero, may appear in thin films if the pair exchange energy for the electrons localized at the magnetic vacancies is positive and large enough at distances suitable for the magnetic ordering between vacancies to emerge. We estimated the exchange energy in the framework of adopted model [42]. It turned out that the pair exchange integral between the vacancies located near the film–substrate boundary z = 0 can be estimated as J ðRÞ≈

 n   Jn γR −γR ⋅  exp ;  aB εi þ εS aB

ð8Þ

2

of factor γ is typically of the order of 0.1 where J n e 2πεe a > 0, the value 2 0 B 0ℏ [45], and aB ¼ ðεi þ εS Þ 2πε is the effective Bohr radius. The parameter 2 μe n = 1 for the lowest wave function 2pz3pz, and n > 1 for higher d- or f-states. The average distance R between the magnetic vacancies depends on their distance z from the film–substrate boundary, z = 0, as follows: RðzÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 3 : πNm V ðzÞ

ð9Þ

Keeping in mind that the vacancies are magnetic only near the surface [42], in what follows we will use the core-and-shell model. In this model, the concentration of magnetic vacancies in Eq. (9)

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M.D. Glinchuk et al. / Thin Solid Films 534 (2013) 685–692

includes the unit-step function θ[zmax − z], where zmax ∝ 3aB⁎. Then, the concentration could be written down as follows: m NV ðzÞ

 0
V 1   β11 þ βV22 um z Aθ½z max −z: exp − ¼ NV0 exp@ hc kB T ðs11 þ s12 Þ

ð10Þ

According to Eqs. (9) and (10), the distance R can be finite only in the layer 0 ≤ z b zmax. For the vacancies located near the film surface, z = h, the quantity (εi + εS) should be substituted by (εi + εe). Since J(R) > 0, Eq. (8) favors the emergence of ferromagnetic spin state irrespective of the distance R between the defects, although the exchange integral is significant at distances R ~ (3–30) aB⁎ and has a pronounced maximum at a distance R ≈ 5 aB⁎ [42].

3. Dimensionless variables and their estimation Let us introduce the dimensionless coordinate ˜z , concentration nV ð˜z Þ, exchange constant jð˜z Þ, and magnetization mð˜z Þ using the formulas ˜z ¼

z z max

;

nV ð˜z Þ ¼

Nm V ð˜z Þ ; NV0

jð˜z Þ ¼

J ðRð˜z ÞÞ ; Jn

mð˜z Þ ¼

Mz ð˜z Þ : μ B NV0

ð14Þ

The characteristic temperature θ, magnetic field H0, misfit parameter uβ, and radius r0 look like
k T θ¼ B ; Jn

k T H0 ¼ B ; μB

uβ ¼

 βV11 þ βV22 um

3kB T ðs11 þ s12 Þ

;

r0 ¼

γ aB

sffiffiffiffiffiffiffiffiffiffiffi 6 3 : πNV0

ð15Þ

2.4. Defect-induced ferromagnetic long-range order in thin films The defect-induced ferromagnetism can have a percolation origin [48], especially in thin films, where the problem dimensionality is reduced down to 2D owing to the spatial confinement. In the continuous medium approximation used in our calculations, the most appropriate is the well-known problem of percolation for spheres, with the magnetic defects being randomly distributed over the sphere centers [43]. At the given temperature T, the radius of percolation for the system of magnetic defects was determined from the condition of equality between the exchange and thermal energies, J(R) = kB T. We analytically calculated the magnetization and the susceptibility of the film for various film thicknesses, temperatures, and vacancy concentrations. On the basis of the statistical physics approach [49], it is possible to derive the following formula describing how the z-dependence of spontaneous magnetization varies with the film thickness and the temperature, 

m μ B NV ðzÞ tanh

Mz ðH; zÞ ¼

  1 MðH; zÞ μ B H þ J ðRðzÞÞ ; m kB T μ B N V ðzÞ

ð11Þ

h

1 ∫ dzMz ðH; zÞ: h0

ð12Þ

Here, H is the magnetic field strength. The exchange constant J(R(z)), the distance R(z), and the concentration of magnetic vacancies NVm(z) in Eqs. (11) are given by Eqs. (8)–(10), respectively. The temperature Tc of ferromagnetic phase transition is determined from the condition J ðRðzÞÞ ¼ kB T c . The saturation magnetization MSAT(z) = μBNVm(z) is determined by the parameter NVm(z) only, being independent of the exchange integral. The corresponding average value depends on the film thickness as follows: h

  mð˜z Þ H jð˜z Þ mð˜z Þ ¼ tanh ; þ nV ð˜z Þ H0 θ nV ð˜z Þ

h

m¼

1 ∫ dzmðzÞ h

ð16Þ

0

where    z θ½1−˜z ; nV ð˜z Þ ¼ exp 3uβ exp −˜z max hc

ð17Þ

   r0 z exp −uβ exp −˜z max εi þ εS h   c z max θ½1−˜z : −r 0 exp −uβ exp −˜z hc

ð18Þ

jð˜z Þ ¼

and to calculate the average value M z ðH; hÞ ¼

Note that the sense of dimensionless parameter uβ consists in the direct product of the elastic dipole and misfit strain magnitudes. Hence, the variation of misfit strain value and sign (e.g. by changing the substrate) will affect the value and sign of uβ. This means that the ferromagnetic properties of thin films can be tuned by the appropriate choice of substrate. Typical ranges of parameters (Eq. (15)) are listed in Table 1. Using the quoted parameters, we can estimate the average ferromagnetic Curie temperature, TC = 〈J(R(z))〉/kB > (400– 800) K. Using definitions (14) and (15), one can rewrite Eq. (11) in the form,

The dimensionless distance between defects equals r ð˜z Þ ¼ γRaðz˜ Þ. B Using the Taylor expansion tanh(x) ≈ x − x 3/3, we can estimate the spontaneous magnetization in the vicinity of transition point jð˜z Þ ¼ θ, 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3   > > jð˜z Þ < n ð˜ Þ 3 θ −1 ; z V mð˜z Þ ¼ jð˜z Þ θ > > : 0; jð˜z Þ<θ:

jð˜z Þ > ˜ θ;

ð19Þ

h

μB μ m ∫dzN V ðzÞ ¼ B ∫dzN V ðzÞθ½z max −z ¼ h h  1  0 0
0 V 0
V 11 8 0   > β11 þ βV22 um β11 þ βV22 um > h h AA > c A−Ei@ > μ B N V0 @Ei@ ; h > < hc h kB T ðs11 þ s12 Þ kB T ðs11 þ s12 Þ  1  0 0
V 0
V 11 ¼ V V   > > β11 þ β22 um > hc @ @ β11 þ β22 um A z max AA > @ > ; h≥z max > : μ B N V0 h Ei k T ðs þ s Þ −Ei k T ðs þ s Þ exp − h B 11 12 B 11 12 c

M SAT ðhÞ ¼

ð13Þ ∞

where EiðzÞ ¼ − ∫ dt expð−t Þ=t is the exponential integral function. In the case when−zthe misfit strain originates from different thermal expansion coefficients for the film and the substrate, i.e. um(T) = V V (αf − αs)(T − Tgrowth), we obtain that the quantity (β11 + β22 ) um(T) linearly depends on the temperature, so that the temperature dependence of magnetization M SAT ðhÞ is weak.

Table 1 Typical ranges of parameters at room temperature T = 300 K.
 qffiffiffiffiffiffiffiffi θ ¼ kJB T e 10−3 –10−1 , r 0 ¼ aγ 3 πN6V0 eð1–10Þ, n

B

    ðβV11 þβV22 Þum   uβ  ¼  3kB T ðs11 þs12 Þ≤10

H 0 ¼ kμB T e400 T B

μB = 927 × 10−26 J/T, ε0 = 8.85 × 10−12 F/m, e = 1.6 × 10−19 C kBT = 0.4 × 10−20 J, kB = 1.3807 × 10−23 J/K, J n e 2πε ea γ10 ~ 10−18–10−19 J, NV0 ~ 1015 2

0 B

−3

cm –1021 m−3 (but NV ≪ 1028 m−3, otherwise approximation (6) is invalid) a⁎B ~ (1–10) nm, γ ~ 0.5, hc ~ (1–10) nm, εi + εS ~ 10

|um| ~ 0.01–0.05, (s11 + s12)−1 ~ 1012 J/m3, |βijVσij(0)| ~ 4 × 10−20 J

M.D. Glinchuk et al. / Thin Solid Films 534 (2013) 685–692

(d) a high spontaneous magnetization, mð˜z Þ≫10, appears only if the misfit parameter uβ is positive; the magnetization is absent if uβ ≤ 0.

Then, the saturation magnetization in terms of dimensionless variables looks like

m SAT

   8
 h > > ; h > : Ei 3uβ −Ei 3uβ exp − max ; h≥z max : hc

ð20Þ

4. Size effects Fig. 2 shows the dependencies of the dimensionless vacancy concentration, average distance between vacancies, exchange energy, and spontaneous magnetization on the dimensionless distance ˜z ¼ z=z max from the film–substrate boundary, z = 0, calculated for various values of misfit parameter uβ. The figure makes it evident that (a) vacancies are accumulated (i.e. the strong inequality nV ð˜z Þ≫1 is satisfied) near the boundary z = 0 if the misfit parameter uβ is positive; if uβ = 0, their concentration is constant across the film and equal to the bulk value; (b) exponential dependences of the vacancy concentration leads to that the average distance between magnetic vacancies, r ð˜z Þ, rapidly changes near the boundary z = 0; in particular, it drastically decreases if uβ > 0; (c) the exchange energy jð˜z Þ increases or has a pronounced maximum in the interval 0≤˜z ≤2 if the misfit parameter uβ is positive; if uβ = 0, the exchange energy jð˜z Þ is constant;

689

One can see from Fig. 2 that all the considered quantities drastically change at z = zmax (magnetic vacancies abruptly disappears at z > zmax; the average distance between magnetic vacancies diverges at z > zmax; and the exchange energy vanishes at z > zmax). The spontaneous magnetization disappears abruptly at z > zmax if the positive misfit parameter uβ is high enough (uβ ≥ 2). Those changes stem from the unit-step function θ[zmax − z] in Eq. (10). The unit-step function arises in the used simple core–shell model with the magnetic interface layer at 0 ≤ z b zmax and the nonmagnetic core at z > zmax. More rigorously, the contribution of p-function vanishes continuously as the distance from the boundary z = 0 into the bulk grows. The described smooth changes are really observed. Note that the negative uβ-values bring about the vacancy accumulation at the free surface z = h, where the misfit is absent (this case is not shown in Fig. 2). The numerical calculations showed that the spontaneous magnetization at uβ b 0 is several (3 to 6) orders of magnitude smaller than that in the case uβ > 0, so that it can hardly be observed experimentally. At the same time, if the surface z = h is rough (or corrugated, as was observed experimentally [6,7,12,13]), the spontaneous magnetization can be strongly enhanced. However, the complicated multi-dimensional problem falls beyond the scope of analytical treatment presented in this paper. Fig. 3a and b shows the dependencies of the dimensionless average spontaneous magnetization m on the dimensionless temperature

b

a 104

10

3

103

0

5

r(z)

nV (z)

1 2 102

2

2

1 10

1

3

0 0.5

1 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

z/zmax

z/zmax

c

d 104

5

3 2

103

1

102

3

0 − uβ=0 1 − uβ=1 2 − uβ=2 3 − uβ=3

0.1 0.05 0.01

m(z)

j(z)

0.5

10

0.2

0.4

0.6

z/zmax

0.8

1 − uβ=1 2 − uβ=2 3 − uβ=3

1

1

0 0

2

1

0

0.2

0.4

0.6

0.8

1

z/zmax

Fig. 2. Dimensionless (a) magnetic vacancy concentration nV ð˜z Þ, (b) average distance between magnetic vacancies r ð˜z Þ, (c) exchange energy jð˜z Þ, and (d) spontaneous magnetization mð˜z Þ as functions of dimensionless distance ˜z ¼ z=z max from the film–substrate boundary calculated for various values of misfit parameter uβ (indicated by the numbers near the curves). The dimensionless radius r0 = 10, the dimensionless temperature θ = 0.005, hc = 2zmax and εi + εS = 10.

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θ and film thickness h/zmax, respectively, calculated for various positive values of misfit parameter uβ. The spontaneous magnetization decreases as 1/h when the variable h/zmax increases. It strongly increases as uβ grows. At the same time, the spontaneous magnetization is absent if uβ ≤ 0. As is seen from Fig. 3a, the critical temperature θc of ferromagnetic state depends on the misfit parameter uβ and increases with its growth: in particular, θc(uβ = 1) ≈ 0.75, θc(uβ = 2) ≈ 3.4, and θc(uβ = 5) ≈ 3.75. Fig. 3b demonstrates that the critical thickness of ferromagnetic state depends on the misfit parameter uβ and also increases with its growth: in particular, hcr(uβ = 1) ≈ 30zmax, hcr(uβ = 2) > 200zmax, and hcr(uβ = 5) > 300zmax at hc = zmax. Fig. 3c shows the dependencies of saturation magnetization m SAT on h/zmax calculated for various values of misfit parameter uβ. The magnetization is a small constant (equal to μBNV0 in dimensional units) if uβ = 0 `. The saturation magnetization strongly increases as uβ grows. If uβ is positive, the saturation magnetization decreases as 1/h for large thicknesses, h/zmax > > 1. This fact is in qualitative agreement with the experiment [3,4] It is seen from Fig. 3d that the saturation magnetization depends on the ratio hc/zmax between the characteristic length of misfit strain relaxation hc and the thickness of magnetic layer zmax (see Eq. (4)). The higher is this ratio, the larger is the saturation value (cf. curves for hc/zmax = 0.2, 0.5, 1, and 2). For the case when the positive misfit strain (uβ > 0) does not relax in the magnetic interlayer (i.e. if hc ≪ zmax) the concentration of magnetic vacancies is approximately the same and rather high across the whole interlayer 0 ≤ z b zmax, and every vacancy contributes to the saturation magnetization. In the opposite case, hc/zmax ≫ 1, the concentration of magnetic vacancies is high only in the interval 0 ≤ z b 3hc, whereas the remaining part of magnetic interface layer, 3hc ≤ z b zmax, is poorly filled with vacancies

and, therefore, does not contribute to the saturation magnetization. In effect, the hc value depends on a lot of factors, including the self-consistent interdependence with the misfit strain. However, it can be tuned by the appropriate choice of substrate. In particular, for the magnetization to be high, it is necessary that the inequalities uβ > 1 and hc/zmax > 3 should be satisfied. The equation jð˜z Þ ¼ θ determines the boundary between the nonmagnetic state and the ferromagnetic phase in the film. Let us regard that the whole film is ferromagnetic if there is a ferromagnetic layer of a finite thickness (i.e. an interface layer), and nonmagnetic otherwise. Fig. 4 exhibits the phase diagram of the such a film in the coordinates dimensionless temperature θ vs. misfit parameter uβ. The ranges of existence for the nonmagnetic and ferromagnetic phases are shown. The temperature interval of ferromagnetic phase increases as uβ grows, until its upper limit reaches the critical temperature θc. At temperatures θ > θc, the film is nonmagnetic at any misfit uβ. The value of θc is determined by the maximum of exchange energy j max ð˜z Þ; at the chosen material dielectric constants, εi + εS = 10, this maximum equals about 0.037 (see Fig. 2c). The value of j max ð˜z Þ can also be determined analytically from Eq. (18),

θc ≡j max ð˜z Þ ¼

1 expð−1Þ max ½x expð−xÞ ¼ : εi þ εS x εi þ εS

ð21Þ

5. Discussion of results and their comparison with experiment In accordance with the experimental results [2–4] concerning the observation, at room temperature, of ferromagnetism in thin textured HfO2, TiO2 and In2O3 films produced by pulsed laser deposition (PLD)

b

a

104 100

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102

1 θc

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Fig. 3. (a) Dependences of dimensionless average spontaneous magnetization m on the dimensionless temperature θ for various misfit parameters uβ (indicated by the numbers near the curves); the reduced film thickness h/hc = 10 and hc = 2zmax. (b) Dependences of m on the dimensionless film thickness h/zmax for various uβ; θ = 0.005. (c) Dependences of dimensionless saturation magnetization m SAT on h/zmax for various uβ; hc/zmax = 2. (d) Dependences of m SAT on h/zmax for various hc/zmax; uβ = 2. The other parameters are the same as in Fig. 2.

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4

Nonmagnetic phase

than that of SnO2, MgO, and ZnO nanosheets and thin films [5–7,12,13]. This observation can be explained in the framework of our model as different orientations of spontaneous magnetization in films and particles. If the ferromagnetic magnetization is directed perpendicularly to the surface, it becomes uniformly oriented due to the film texture in thin films. In every nanoparticle, the local c-axis is directed radially. Therefore, the average value of spontaneous magnetization for an ensemble of randomly oriented spherical nanoparticles is very small. As a result, a nonzero magnetization can be a consequence of nanoparticle shape deviation from the sphere. A typical fraction of such particles in the region concerned varies from one to a few percent (see, e.g., Ref. [14]).

θc

θ×102

3

2

1

Ferromagnetic phase 0 0

0.5

1

1.5

2

2.5

691

3

uβ Fig. 4. Phase diagram of the film in the coordinates “dimensionless temperature θ-dimensionless misfit parameter uβ” showing the ranges of existence for the nonmagnetic and ferromagnetic phases. The film thickness h = 10zmax , the critical thickness of dislocations appearance hc = 2zmax, εi + εS = 10; the other parameters are the same as in Fig. 2.

technique at 1150 °C, we conclude that our consideration can explain the following observations. 1) Annealing in the oxygen atmosphere (for a few hours) reduces and afterward (in 8 to 10 h) completely vanishes the magnetic moments of the films, transforming the latter into the diamagnetic state, which proves that oxygen vacancies are the main sources of film magnetism. This statement [2–4] is in complete agreement with the main prediction of our quantum-mechanical model (see Section 2.3) concerning the magnetic state of oxygen vacancies in the vicinity of the surface; namely, two electrons, a 2p and a 3p one, localized at the defect are responsible for the vacancy spin s = 1. 2) The origin of magnetism is the same in all the films concerned, with the number of oxygen vacancies being independent on the film thickness. On the other hand, the film magnetization is strongly anisotropic and depends on the substrate type, texture, and orientation. Therefore, the oxygen vacancies seem to be mostly located at the film–substrate interface [2–4]. This statement is in complete agreement with the result obtained in the framework of our model that the oxygen vacancies are accumulated in the vicinity of the film–substrate boundary. The accumulation occurs owing to the interaction between the vacancy elastic dipoles and a non-uniform strain field. The latter emerges from misfit strains, which, as a rule, result from a mismatch between the film and substrate lattice constants and the difference between their thermal expansion coefficients. Thus, the vacancy accumulation is substrate-dependent. At the same time, we managed to explain how misfit strains can reorient the crystallographic axes and create textures in thin polycrystalline films (see Section 2.3). 3) The saturation magnetization is almost temperature-independent up to 400 K. Moreover, it does not depend on the film thickness “in a simple way”, but strongly increases as the thickness decreases (from about 0.03 10 6 A/m for a 200-nm HfO2 film to more than 0.4 10 6 A/m for a 10-nm film) [3,4]. The results of our calculations in Section 4 (see Fig. 3c–d) are in qualitative agreement with this observation. The calculations of spontaneous magnetization and ferromagnetic transition temperature also showed that the model proposed describes the experimental results quite well. 4) The saturation magnetization of SnO2, MgO, and ZnO nanoparticles was experimentally found to be much (up to 100 times) smaller

In accordance with the experimental results [6,7,12,13] concerning the room-temperature ferromagnetism in thin polycrystalline MgO and ZnO films produced by magnetron sputtering at room temperature in an atmosphere with various partial pressures of oxygen in the chamber during the deposition process, we draw the following conclusions. 1) The saturation magnetization MSAT of the films deposited onto different substrates has the same behavior. Since the films were deposited at room temperature, misfit strains, which arise in the polycrystalline film owing to the difference between the thermal expansion coefficients for the film and the substrate, are most likely absent, so that there is no strain-induced accumulation of defects at the film–substrate interface. As a result, the magnetization is not sensitive to the substrate choice [13]. However, owing to a low mobility of atoms at room temperature, a noticeable roughness appears on the oxide surfaces in MgO/Pt and ZnO/Pt heterostructures. This roughness enlarges the effective surface area, although the film–substrate boundary looks like smooth. Supposing that the MSAT value for films deposited at 350 °C is significantly reduced in comparison with that for films deposited at room temperature [13] (this may take place, e.g., because the atomic mobility increases at higher temperatures and, as a result, the surface roughness decreases [50]), we arrive at the conclusion that the top surface could play a decisive role in the emergence of film ferromagnetism. In particular, it can be induced by magnetic oxygen vacancies located in the vicinity of this surface rather than by other magnetic defects, the concentration of which, according to estimates of Ref. [13], is too low for the magnetic percolation between them to appear. 2) The saturation magnetization of MgO films on the Si substrate non-monotonically depends on the partial pressure of oxygen; namely, it has a sharp maximum at a pressure of about 13 mPa and then decreases as the pressure increases (see Fig. 5a). If the oxygen pressure exceeds 300 mPa, the films become diamagnetic. The decrease of saturation magnetization as the oxygen pressure increases can be explained on the basis of the fact than the concentration of oxygen vacancies decreases with the pressure increase at high pressures (this fact was established for other materials as well, e.g., for La1 − xSrxCoO3 − y [51]). The exact dependence of vacancy concentration in MgO on the partial oxygen pressure is unknown for low pressures, and we cannot explain a sharp maximum experimentally observed at 13 mPa. However, as is seen
 ðp=p0 Þ from Fig. 5a, the asymptotic dependence MSAT ðpÞ∝ 1− lnln ðp max =p0 Þ approximates the experimental points at high pressures, p ≫ p0. 3) The saturation magnetization of MgO and ZnO films on different substrates depends non-monotonically on the film thickness. In particular, it has a sharp maximum at a thickness of 170 or 250 nm, respectively, and decreases if the film thickness grows further (see Refs. [12,13] and Fig. 5b). In Fig. 5b, we compare the theoretical and experimental dependences of saturation magnetization of the film thickness obtained for MgO/Si films. One can see that the asymptotic dependence MSAT(h) ∝ 1/h (see Section 4, Fig. 3d) approximates the experimental points at large thickness values. Nevertheless, additional studies, both theoretical and

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b 1.75

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0 0

50 100 150 200 250 300

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0

100

200

300

400

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Film thickness h (nm)

Fig. 5. Dependences of saturation magnetization MSAT on (a) the oxygen pressure p at a fixed film thickness h and (b) on the film thickness h
at a fixed oxygen pressure p. Symbols  ðp=p0 Þ correspond to experimental data for MgO films on Si substrate taken from Ref. [12]. Solid curves are the fitting approximations (a) M SAT ðpÞe 1− lnln with p0 = 10 mPa and ðp max =p0 Þ pmax = 300 mPa, and (b) MSAT(h) ∝ 1/h.

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