Exploring the 2D to 3D dimensionality crossover in thin iron films

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Journal of Magnetism and Magnetic Materials 300 (2006) 519–524 www.elsevier.com/locate/jmmm

Exploring the 2D to 3D dimensionality crossover in thin iron films U. Ko¨bler, R. Schreiber Forschungszentrum Ju¨lich, Institut fu¨r Festko¨rperforschung, D-52425 Ju¨lich, Germany Received 29 August 2005 Available online 25 January 2006

Abstract The temperature dependence of the spontaneous magnetization of epitaxial iron films with a thickness ranging from d ¼ 20 to 200 nm has been measured. The films are grown on GaAs (1 0 0) substrates which are covered by a 150 nm thick silver (1 0 0) buffer layer. For three-dimensional BCC iron it was observed already in 1929 that saturation of the spontaneous magnetization for T ! 0 is perfectly described by a T2 power law. On the other hand, for thin two-dimensional (2D) iron films a T3/2 law has been established in many recent experimental investigations. In our iron films grown on diamagnetic silver, this dimensionality change occurs at a thickness between d ¼ 100 and 200 nm. Comparison of the here-observed T3/2 coefficients with those on iron films grown on paramagnetic tungsten (1 1 0) shows that the 2D interactions are
20 times larger in the films on tungsten. Recent results on Fe films which are grown directly on GaAs (1 0 0) confirm that the substrate has a very strong effect on the coefficient of the T3/2 function, i.e. on the strength of the magnetic interactions in the films. r 2006 Elsevier B.V. All rights reserved. PACS: 75.70.Ak Keywords: Ferromagnetic order parameter; Dimensionality crossover

1. Introduction There is still a considerable confusion concerning the analytical function for the temperature dependence of the spontaneous magnetization, MS(T), of BCC bulk iron. In 1929, Weiss and Forrer measured MS(T) in the range 90oTo290 K with high precision and observed that MS(T) is perfectly described by a T2 function [1]. In this temperature interval, MS(T)/MS(T ¼ 0) decreases from 0.998 to 0.983. Later on, Butler et al. [2] performed spontaneous magnetization measurements in an extended temperature range of 77oTo398 K. These data confirm the T2 dependence of Ref. [1]. Fig. 1 reproduces the normalized two data sets on a T2 scale in order to demonstrate their agreement and the quality of the T2 description. Also the data of Crangle and Goodman [3] are consistent with a T2 power law over a temperature range of up to 467 K. As a conclusion, for the investigated Corresponding author. Tel.: +49 2461 613324; fax: +49 2461 612610.

E-mail address: [email protected] (U. Ko¨bler). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.003

temperature range the assumption of an exact T2 function is consistent with the experimental results. It should be noted that there is a specific experimental problem with accurate low-temperature magnetization measurements. In these macroscopic measurements, a finite magnetic field has to be applied in order to saturate the sample magnetically. This field must be slightly larger than the demagnetization field which is BD ffi 0:73 T for the iron sphere. Paramagnetic impurities from the sample support can, therefore, add a perturbing
T1 dependence to the intrinsic magnetic signal. These contributions can be dominant at very low temperatures where the temperature dependence of spontaneous magnetization is extremely small. This has the effect of reducing the fitted exponent below the value of e ¼ 2. In unfavorable cases, the macroscopic spontaneous magnetization data for T ! 0 can deviate visibly towards larger values with respect to the T2 function observed at higher temperatures. This is the case for the iron data but not for the nickel data of Ref. [4]. Also in Ref. [5] paramagnetic contaminations seem not to have been properly eliminated.

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520

1.00

bulk iron P.Weiss, R. Forrer Ann. Physique 12 (1929) 20

Table 1 Universal Te power functions of the magnetic order parameter at the relevant interactions and of the spin quantum number Exchange interactions

M / M (T=0)

0.99

Integer spin

0.98

Half-integer spin

1D 2D Anisotropic

2D 3D Anisotropic

T3

T2

5

T2

3

T2

3D 9

T2 T2

For anisotropic 3D interactions and isotropic 2D interactions (second line) the exponents are identical. The same applies for anisotropic 2D interactions and 1D interactions.

0.97 M.A. Butler, G.K. Wertheim, D.N.E. Buchanan Phys.Rev. B 5 (1972) 990 0.96 0

20

40

60

80

100

120

140

160

T2 / 1000 (K2) Fig. 1. Normalized macroscopic spontaneous magnetization of BCC bulk iron after Refs. [1,2] vs. square of absolute temperature. These data can empirically be described by a single T2 function. According to Table 1 this conforms to the universality class with half-integer spin (S ¼ 32) and isotropic interactions. Also data from Ref. [3] confirm the T2 dependence.

Those problems are avoided in zero field methods such as 57Fe NMR, Mossbauer spectroscopy or neutron scattering. In Ref. [6] we have fitted a single power function to the 57Fe NMR data of BCC iron from Ref. [2] and obtained for the exponent e ¼ 2:01  0:02. The exponent of e ¼ 2 was verified for many other insulating and metallic magnets with half-integer spins and isotropic interactions. The low-temperature T2 function reported for the first time for iron in 1929 is not understood theoretically up to now. This concerns not only the value of the exponent but also the fact that the T2 function holds exactly over a large temperature range. As we know from renormalization group theory, T ¼ 0 is a second stable fixed point in addition to the critical temperature Tc [7]. In the vicinity of those points, simple power functions with universal exponents hold exactly over a considerable temperature range. Universality at T ¼ 0 results from the very large length scale of the dynamics in the long-range ordered state. As a consequence of this quasi-macroscopic length scale, all microscopic details such as spin structure and lattice structure are averaged out and not important for the dynamics. Note that the characteristic symmetry is invariance with respect to transformations of the length scale. This is the symmetry of a continuum. For the insulating transition metal compounds with localized magnetic moments and well defined spin quantum numbers, an empirical scheme of six universal exponents e could be established for the saturation of the order parameter for T ! 0 [6,8]. These exponents apply to the magnetic excitations such as magnon energy gaps and stiffness constants as well [9]. We reproduce this scheme in Table 1 in order to illustrate how the exponent e depends on the dimensionality of the relevant interactions and on the spin quantum number.

It is very surprising that different universality classes hold for integer and half-integer spin values. Considering that the number of states, N ¼ 2S þ 1, is the most important quantity for the dynamics, an integer spin is equivalent to an odd number of states and a half-integer to an even number of states per magnetic particle. A spin-dependent dynamics is not known for the paramagnetic phase. For instance, the T1 dependence of the Curie–Weiss susceptibility holds for all spin values. Note that in contrast to the long-range ordered state the length scale of the dynamics in the paramagnetic phase is given by the near-neighbour distance, i.e. the atomistic lattice structure sets a short wavelength limit to the high temperature magnetic fluctuations. Under this condition, the short-distance Heisenberg interactions between a small number of neighbours are the relevant interactions. These interactions are identical for integer and half-integer spin values. In the long-range ordered state, the situation is completely different. As is well known, the correlation length becomes infinitely long at the ordering transition. The length scale of the dynamics in the long-range ordered state is, practically, infinite. As a consequence, new types of long-distance interactions can become relevant? Classical dipole–dipole interaction is the only known long-distance interaction but evidently not able to explain a different dynamics for integer and half-integer spins. Speaking in terms of renormalization group theory dipole–dipole interaction does not seem to be the relevant interaction. As a conclusion, a hitherto unknown type of long-distance interaction has to be invoked in order to explain different universality classes for integer and half-integer spins. In Ref. [10] arguments were given that the associated magnetic excitations are magneto-elastic modes. These excitations are difficult to observe because they have very long wavelengths and very low energies. The universality classes for T ! 0 are represented by the power functions of Table 1. As a consequence, power series expansions for the temperature dependence of the spontaneous magnetization according to conventional spin wave theory [11] must be viewed as a listing of different universality classes. It is evident that this is physically not

ARTICLE IN PRESS U. Ko¨bler, R. Schreiber / Journal of Magnetism and Magnetic Materials 300 (2006) 519–524

that the number of states can only increase with increasing temperature. This interpretation is in analogy to the crossover from low temperature T2 to a high temperature T9/2 function observed at about T n ¼ 628 K in the spontaneous magnetization of BCC bulk iron [13]. Since iron is cubic for all temperatures the interactions can be assumed to be 3D isotropic throughout. It is, therefore, safe to interpret the T2 to T9/2 crossover as a change from a half-integer (S ¼ 3=2) to an integer (S ¼ 2) spin quantum number, i.e. as an increase of the number of relevant states from N ¼ 4 to N ¼ 5. The aim of this communication is to investigate the 2D to 3D dimensionality change of epitaxial iron films as a function of the film thickness using macroscopic magnetization measurements. Our iron films are grown on a silver (1 0 0) buffer layer. Comparison with iron films on tungsten [16] and with recent results on iron films which are grown directly on GaAs (1 0 0) [28] shows that the substrate has a very strong effect on the strength of the 2D interactions in the films. 2. Experimental Prior to discussing the new experimental results of this work, we outline our method for the identification of the 2D to 3D dimensionality crossover on account of the 57Fe Mo¨ssbauer data obtained on ultrathin iron (1 1 0) films on tungsten (1 1 0) after Ref. [16]. Fig. 2 shows the temperature dependence of the normalized hyperfine fields from Ref. [16] on a T3/2 scale. It can be seen that for all iron thicknesses the thermal decrease of the hyperfine field, i.e. the spontaneous magnetization, follows T3/2 dependence in the temperature interval shown. This analytically simple

100

T (K) 300

200

400

1.00 iron 20.5

0.95 Hhf (T) / Hhf (T=0)

reasonable since only one universality class can hold at a time. The fact that the exponents of Table 1 are observed also for the itinerant ferromagnets Ni, Fe and Co [12,13] suggests that an effective spin seems to be defined also for these materials. In particular, the exponent of e ¼ 2 observed for BCC iron with isotropic 3D interactions unambiguously shows that the spin is half-integer. Considering that the saturation moment is 2.217 mB per iron atom [4,14] it is reasonable to assume that the spin S eff ¼ 3=2, i.e. there is a fourfold ground state of the iron atom. Table 1 shows furthermore that the T3/2 law observed for the ultrathin crystalline iron films [15–17] is characteristic for a half-integer spin (S ¼ 3=2) and isotropic 2D interactions. On the other hand, T3/2 laws were observed also for many amorphous iron and iron–nickel alloys such as Metglass 2826 with the composition Fe40Ni40P14B6 [18–25]. In these systems the interactions are 3D but locally anisotropic. Moreover, T3/2 laws are observed also in noncubic crystalline solids such as the hexagonal ferromagnets gadolinium [26,27] and cobalt [13] in which the relevant interactions are 3D and (macroscopically) anisotropic. This shows that physically different systems can fall into the same thermodynamic universality class. These universality classes are stable within due limits against various nonrelevant parameters. Crossover to another universality class is induced only if these parameters exceed a threshold value. In Ref. [12] we have analyzed 57Fe Mo¨ssbauer spectroscopic data of Ref. [16] measured on ultrathin crystalline iron (1 1 0) films which were grown epitaxially directly on tungsten (1 1 0). For these iron films, a crossover in the hyperfine field (which is assumed to be proportional to the spontaneous magnetization) from a low-temperature T3/2 function to a high temperature T2 function could be identified. Interpretation of this crossover is ambiguous since the T2 function pertains to 2D systems with integer spin and to 3D systems with half-integer spin as well. Note that the dimensionality of the relevant interactions can only decrease with decreasing temperature. The exponent e, therefore, can change only from right to left in Table 1 as a function of decreasing temperature. Crossover from hightemperature T2 dependence to low-temperature T3/2 dependence, i.e. from 3D to 2D interactions is, therefore, consistent with Table 1. On the other hand, interpretation of the T3/2 to T2 crossover as a dimensionality change from 2D to 3D is physically not plausible for ultrathin iron films. This crossover was identified for all iron films on tungsten with more than three atomic layers. Three atomic layers seems to be much too less for the realization of 3D behaviour. It is, therefore, more likely that the films remain 2D and that the number of relevant states increases from N ¼ 4 (Seff ¼ 3=2) to N ¼ 5 (Seff ¼ 2) at the crossover to the high temperature T2 function. Note

521

8.6 0.90

5.3 atomic layers : 3.4

0.85

J. Korecki, M. Przybylski, U. Gradmann JMMM 89 ( 1990 ) 325

0.80 0

2000

4000 T

6000 3/2

(K

8000

10000

3/2)

Fig. 2. Normalized 57Fe hyperfine field data (main spectral component) of epitaxial ultrathin iron (1 1 0) films grown on tungsten (1 1 0) obtained by Mo¨ssbauer spectroscopy after Ref. [16] vs. absolute temperature to the power of 32. According to Table 1 the T3/2 law pertains to the universality class with isotropic 2D interactions and half-integer spin (S ¼ 32).

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[M (T=0)-M(300K)]/M (T=0)

0.12

Fe (110) on tungsten (110)

0.10 0.08 0.06 J.Korecki, M.Przybylski, U.Gradmann 0.04

JMMM 89 (1990) 325

0.02

bcc bulk iron

0.00 0.0

0.5

1.0 d-2

1.5

2.0

(nm-2)

Fig. 3. Relative decrease of the 57Fe hyperfine field (spontaneous magnetization) between T ¼ 0 and T ¼ 300 K vs. inverse film thickness squared (d2). An excellent linear dependence results in this diagram. The d2 dependence is interpreted as due to the two surfaces of the films. Extrapolation to d ! 1 gives a larger value for the 2D films compared to bulk iron.

2.20 (µB /Fe)

behaviour allows us to calculate the relative decrease of the spontaneous magnetization DmðTÞ ¼ ½MðT ¼ 0Þ  MðTÞ=MðT ¼ 0Þ between T ¼ 0 and, say, T ¼ 300 K by fitting a T3/2 function to the experimental points. Fig. 3 displays this decrease of the normalized magnetization Dm(300 K) vs. the reciprocal thickness of the Fe films squared, d2. An excellent linear behaviour results in this plot. We interpret the d2 dependence as a consequence of the necessary condition that a 2D film has two boundaries with each boundary contributing a factor of d1. Apparently, all other parameters of the films are fairly independent of thickness. This is confirmed by the fact that all films exhibit saturation hyperfine fields for T ! 0 which are very close to the bulk value (Bhf ¼ 33:95 T). We should note that the Mo¨ssbauer spectra of the ultrathin films contain two spectral components and that we discuss here the high-field hyperfine component which is characteristic for all layers except the first one on the tungsten substrate [16]. It is no surprise that for the 2D films the extrapolated Dm(300 K) value for d ! 1 is larger than the bulk value. This is because the spontaneous magnetization of the films decreases with a larger rate Dm(T) than the spontaneous magnetization of bulk iron. Fig. 3 demonstrates that crossover to bulk properties is not simply obtained by the transition d ! 1 but is associated with a functional change. Note that we have calculated Dm(300 K) for bulk iron from the T2 fit function through the data of Fig. 1. The 2D to 3D crossover is apparent from Fig. 3 but it is not resolved because it occurs at a much larger thickness. Comparison with our iron films on silver reveals that Dm(300 K) is considerably larger for the films on silver compared to films with the same thickness on tungsten. In Fig. 4, we show absolute values for the spontaneous

Fe (100) 28nm on silver 2.18 ~T2

2.16 2.14 2.12 m/Fe

522

2.10 2.08 0

1000

2000 T3/2

3000

4000

5000

(K3/2)

Fig. 4. Absolute macroscopic spontaneous magnetization of an iron (1 0 0) film with a thickness of d ¼ 28 nm (
195 atomic layers) grown on a 150 nm silver (1 0 0) buffer layer vs. T3/2. The perfect T3/2 law reveals that the relevant interactions are 2D in the temperature range shown. Note that for bulk iron the saturation moment is 2.217 mB per iron atom. A T2 function would be curved as indicated on the T3/2 scale.

magnetization of an iron (1 0 0) film on silver with d ¼ 28 nm (
195 atomic layers). These data which have been obtained with a Faraday balance magnetometer confirm the T3/2 function with a high precision. If these data would follow T2 dependence the indicated curved behaviour would result on the T3/2 scale. From the data of Fig. 4 we calculate Dm(300 K) ¼ 0.0512. Such a Dm(300 K) value pertains to a thickness of 1.75 nm (
8.6 atomic layers) for the iron films on tungsten according to Fig. 3. As a result, the iron film on tungsten with d ¼ 1:75 nm has properties like an iron film on silver with d ¼ 28 nm. It seems appropriate to define an effective thickness for the films on tungsten which is much larger than the physical thickness. Tungsten as a paramagnetic substrate seems to support the magnetic interactions and lets the iron films appear thicker. We can assume that with the larger effective thickness the ordering temperature is also larger. In Fig. 5 we have plotted our Dm(300 K) data for the iron films on silver on a d2 scale. Like for the corresponding diagram for the iron films on tungsten (see Fig. 3) a fairly linear dependence results for the films with dp100 nm. However, the slope is about a factor of 430 larger. For d ! 1 a value of Dm(300 K) ¼ 0.0297 results by linear extrapolation which is again larger than the value for bulk iron of Dm(300 K) ¼ 0.020. Note that the ratio of these two Dm(300 K) values scales nicely with the ratio of the dimensionalities of 2/3. The iron film with thickness d ¼ 200 nm shows T2 dependence and, therefore, is 3D. Fig. 6 displays the normalized spontaneous magnetization of this film on a T2 scale. If the data would follow T3/2 dependence the indicated curved behaviour would result on the T2 scale.

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0.065 0.060

d=22.5nm

[M (0)-M (300 K)]/M (0)

0.055 0.050

d=28nm

0.045 0.040

d=100nm

d=40nm

0.035 d=60nm

0.030 0.025

d=200nm

0.020

Fe thin films on silver

bcc bulk iron 0.015 0.0000

0.0005

0.0010

d-2

0.0015

0.0020

(nm-2)

Fig. 5. Relative decrease of spontaneous magnetization in the range 0oTo300 K for the iron films on silver vs. reciprocal film thickness squared (d2). Compared with the corresponding data for ultrathin iron films on tungsten (see Fig. 3) a 430 times larger slope is observed for the films with dp100 nm. The extrapolated value for d ! 1 of 0.0297 is clearly larger than the value for BCC bulk iron of 0.020. Crossover from 2D to 3D behaviour is between d ¼ 100 nm and d ¼ 200 nm. Note that the films with dp100 nm show T3/2 dependence but the film with d ¼ 200 nm T2 dependence.

1.000

iron d=200 nm

0.998

MS/MS (T=0)

0.996 0.994 0.992 0.990 0.988 ~T3/2

0.986 0.984 0.982 0.980 0

20000

40000

60000

80000

T2 (K2) Fig. 6. Temperature dependence of normalized spontaneous magnetization of the iron (1 0 0) film with d ¼ 200 nm grown on silver vs. square of absolute temperature. The T2 dependence shows that this film is 3D. A T3/2 function would be curved as indicated on the T2 scale.

Note that for this film the Dm(300 K) value is very close to the bulk value according to Fig. 5. 3. Conclusions We have investigated the 2D to 3D dimensionality change in thin iron films using the empirical universality classes of Table 1 as a guide. Originally these universality classes were established for insulators but they hold for the itinerant ferromagnets Fe, Ni and Co as well [12,13]. The reason for this seems to be that the number of relevant states per magnetic particle, N, is always a well defined

523

integer. This is essential for the existence of discrete universality classes according to Table 1. For iron it is reasonable to assume that N ¼ 4 (S eff ¼ 3=2) for T ! 0. As a consequence, isotropic 2D iron films fall in the T3/2 universality class and iron films which are sufficiently thick to be 3D fall in the T2 universality class according to Table 1. This dimensionality crossover occurs at a film thickness between 100 and 200 nm for the epitaxial-iron films grown on Ag(1 0 0). The 2D to 3D dimensionality crossover seems to be essentially a matter of the thickness of the film. Note that the saturation hyperfine fields as well as the saturation magnetic moments observed for the films are identical with the bulk values. No significantly enhanced magnetocrystalline anisotropy is observed for the epitaxial-iron films compared to bulk iron. On reducing the film thickness, all spin waves which propagate normal to the film plane and which have a wavelength longer than the thickness of the film are cut off. For the iron films on silver, we have observed that the 3D to 2D crossover occurs if spin waves having a wavelength larger than 100 nm are cut. This shows that the relevant spin waves in the 3D long-range ordered state have wave vectors qo0.063 nm1. Inelastic neutron scattering experiments have normally not been conducted to such small wave vector values [29,30]. The reason for this is the limited energy resolution of inelastic neutron scattering which impedes detection of the associated small excitation energies. Note that in the ordered state the long-range interactions for q ! 0 are weak but they couple all spins of the sample. As a consequence, we have to conclude that the experimentally observed magnetic excitations are not necessarily characteristic for the long-range ordered state because they belong to too large wave vector values. In fact, the observed spin wave dispersion curves are well described by the strong short-distance Heisenberg interactions which act between a small number of neighbours only. It is important to note that there is a considerable inconsistency between the magnetic excitation spectra which can be described by the short-distance Heisenberg interactions and the observed universality of the order parameter for T ! 0 according to Table 1. In fact, the excitation spectra exhibit, evidently, no universal features and are material specific. For instance, the spin order type belongs to the material specific features on the length scale of the lattice parameter. As is well known, the magnon dispersion curves of ferromagnets start as a quadratic function of q but for antiferromagnets as a linear function of q. Nevertheless the two types of magnets belong to the same universality class for T ! 0 and for T ! T c as well. The reason for this is the large length scale of the dynamics in the ordered state whereby all local details are averaged out. On the other hand, the non-universal material specific properties are given by the atomistic details on the length scale of the crystallographic unit cell.

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Observation of universality for the order parameter clearly indicates that the non-universal short distance interactions of the excitation spectra are not the relevant ones for the long-range ordered state. It appears that inelastic neutron scattering samples local excitations only. In inelastic processes, the particle character of the neutron seems to be more important than the wave character. In the 2D films, only a limited number of discrete spin waves perpendicular to the film exist. If these spin waves are pinned to the surface they give rise to rather sharp Brillouin light scattering signals [31]. Because the number of the discrete perpendicular spin waves is much smaller than the in plane spin waves, they are not relevant for the dynamics. As a consequence of the 2D order there are only two types of domains with magnetization vectors in the film plane. Comparison of the coefficients of the T3/2 functions for the iron films on silver (1 0 0), on tungsten (1 1 0) [16] and on GaAs (1 0 0) [28] shows a tremendous influence of the substrate. Since for all three systems, the thickness dependence of the coefficient of the T3/2 function is given to a good approximation by d2 dependence the coefficient of the d2 function provides a convenient means to demonstrate the effect of the substrate. According to Figs. 3 and 5 this coefficient is 0.039 nm2 for the tungsten (1 1 0) substrate and 16.8 nm2 for the silver (1 0 0) substrate. A similar analysis of the data of Ref. [28] for iron on GaAs (1 0 0) gives
0.06 nm2. Note that the interaction decrease is stronger upon film thickness decrease if this coefficient is larger. In particular, iron films on tungsten have properties as iron films on silver with a
20 times larger thickness. As a possible reason for the different d2 coefficients we propose that paramagnetic tungsten supports the magnetic interactions but diamagnetic silver weakens the interactions. In any case metallic substrates seem to affect the 2D interactions strongly. Perhaps the non-metallic GaAs best approaches a magnetically neutral substrate. Of course, the growth modes on the different substrates can have a nonnegligible influence. Acknowledgments We are much indebted to U. Gradmann of TUClausthal-Zellerfeld for many stimulating discussions.

Helpful discussions with M. Buchmeier of FZ-Ju¨lich are also gratefully acknowledged.

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