Magnetization of 3He layers in ferromagnetic regime: Cluster size effects

Physica B 407 (2012) 3925–3932

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Magnetization of 3He layers in ferromagnetic regime: Cluster size effects I.I. Poltavsky, T.N. Antsygina, M.I. Poltavskaya, K.A. Chishko n B. Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkov, Ukraine

a r t i c l e i n f o

abstract

Article history: Received 13 January 2012 Received in revised form 27 April 2012 Accepted 1 June 2012 Available online 9 June 2012

Magnetization of pure 3He monolayers on graphite and solid 3He on 4He-preplated graphite in the ferromagnetic regime is investigated theoretically within a two-dimensional spin-1/2 Heisenberg model in an external magnetic field. We develop an analytical approach based on a second-order twotime Green function formalism which describes the thermodynamic functions for finite as well as for infinite spin systems in the whole temperature range at arbitrary fields h. In particular, the present theory provides a proper description of the magnetization in the intermediate temperature region h o T o J at h well below the exchange J, that is in the region where the measurements for solid 3He monolayers are usually made. It is known from the experiment that only a part of 3He atoms is involved into ferromagnetic exchange. The ferromagnetic spins form clusters whose size is a function of coverage. The coverage dependences of the exchange constant, saturation magnetization, and average cluster size are found and analyzed. Nonmonotonic variation of the exchange constant with coverage is discussed. The effect of the cluster size on the temperature dependence of the magnetization is studied. To clarify how the cluster shape and boundary conditions affect magnetic properties of nanoclusters we calculate numerically the magnetization of small magnetic systems using the exact diagonalization method. & 2012 Elsevier B.V. All rights reserved.

Keywords: Heisenberg model Solid 3He monolayers Ferromagnetic nanoclusters

1. Introduction Magnetic properties of solid 3He monolayers on substrates of various types are extensively investigated both experimentally [1–22] and theoretically [21–30]. These systems are produced by adsorption of 3He atoms directly on exfoliated graphite or on preplated (e.g., with 4He [8,19] or HD [16]) graphite. The second solid 3He monolayer provides an excellent example of a nearly perfect 1/2-spin nuclear magnet on a triangular lattice. Its thermodynamics can be described by the multiple-spin exchange (MSE) model [21,22,29] with cyclic n-particle exchange constants depending on coverage r. With increase in r the 3He monolayer evolves from antiferromagnetic to ferromagnetic behavior. As the second monolayer densifies, the three-spin permutations become predominate, and the ferromagnetic Heisenberg (HFM) model with exchange J of order 1–2 mK is quite appropriate to describe adequately magnetic properties of the 2D 3He [5,21,29,30]. At present a great variety of experimental data on temperature dependence of the magnetization M(T) are available from the literature. Our prime interest is magnetic properties of 2D 3He at dense coverages when the HFM model can be applied. While

n

Corresponding author. Tel.: þ380 97 357 0730; fax: þ380 57 345 0593. E-mail address: [email protected] (K.A. Chishko).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.06.001

interpreting theoretically the experimental data on magnetization the following points should be taken into account. In all cases, magnetization measurements are made in an external magnetic field with typical values B satisfying the condition h=J 5 1 (h ¼ 2mB, where m is the nuclear magneton). Depending on the coverage the solid 3He monolayer behaves either as a homogeneous magnet or as a set of finite-sized ferromagnetic regions (clusters). In the latter case at small external fields the magnetization essentially depends on the cluster size. Solid 3He on graphite preplated by 4He is another example of a clustered system [8,11]. At rather high content of 4He, the magnetic subsystem consists of ferromagnetic 3He inclusions (clusters) in the 2D 4He matrix. The change in 4He concentration regulates the size of 3He clusters. The average size of inclusions varies from tens of atoms (nanoclusters) to large domains (islands) containing hundreds of atoms. From the above it follows that a theory describing the experimental data on M(T) should be built so as to take properly into account both cluster structure of the system and the condition h=J 5 1. Theoretical approaches to date have certain limitations relative to these criteria. Applicability of high-temperature expansions (HTSE) [23,26] is restricted by minimal temperature T  J when the magnetization of 3He is far from saturation. In addition, HTSE is constructed for infinite systems, so that it is not appropriate for finite clusters. Alternatively, the Kopietz formula for the magnetization [24] obtained in the spin-wave approximation is

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valid for a finite 2D system. However, its applicability is restricted by very low temperatures where the magnetization is close to saturation. This means that both these approaches fail to explain the behavior of the system in the sufficiently wide intermediate temperature region, h o T oJ, where the magnetization measurements are usually made. Different numerical methods such as quantum Monte-Carlo simulations (QMC) or exact diagonalization (ED) give results for a wide temperature range, but they both operate with rather small systems and are not appropriate for clusters containing thousands of particles. Moreover, to make numerical calculations it is necessary to specify the parameters of the system. However, a number of these parameters cannot be deduced directly from the experiment and are determined only by fitting the theory to the experimental data. Thus, any analytical approach which yields a way for estimation of all the system parameters is of great importance. In the present paper we develop the theoretical description of the HFM in an external magnetic field with the aim to interpret the experimentally observed magnetic properties of ferromagnetic 3He monolayers on graphite and 3He ferromagnetic clusters on substrates preplated by 4He. We use the second-order Green function formalism based on the decoupling of higher Green functions at the second step with introducing vertex parameters to be found. The scheme was originally proposed in Ref. [31] for a linear spin chain in zero magnetic field and later was extended on 2D Heisenberg magnets [32–36]. In Ref. [37] the second-order Green function formalism was first applied to HFM chain and HFM on a square lattice at nonzero magnetic field. The decoupling scheme [37] was further developed in Ref. [38] to yield more accurate description for the thermodynamics of 1D and 2D HFM (on square and triangular lattices) especially in the low-field region. As mentioned above, it is just this field range that is used in the experiments on 3He monolayers. Here we employ a new decoupling scheme that further improves the description of the HFM thermodynamics in the temperature region ho T o J at ultralow magnetic fields h=J 5 1. The paper is organized as follows. In Section 2, we outline the proposed theoretical method. The self-consistent set of equations for the spin–spin correlators, magnetization, and vertex parameters is derived. In Section 3, the developed scheme is used to interpret the experimental data on the magnetization of pure 3He on graphite and 3He on 4He-preplated graphite. Some concluding remarks are made in Section 4.

The Heisenberg Hamiltonian is given by X X Sf Sf þ d h Szf , H ¼ J

Si Sj Sl

ia j a l,

i al,

ð2Þ z

where a? and az are the vertex parameters, Z ¼ /S S, and angular brackets mean the thermodynamic averaging. While decoupling the last spin combination we represent Szi as Szi ¼ Z þ dSzi and introduce the vertex parameter only in the term containing /dSzi dSzj S. After a number of calculations we obtain for the time–space Fourier component //dSzk 9dSzk SSo //dSzk 9dSzk SSo ¼

Jc1 g0 1Gk , 2p o2 ðoz Þ2 k

ð3Þ

where ðozk Þ2 ¼ 2J2 g0 ð1Gk Þ½Dz þ g0 c~ 1 ð1Gk Þ,

ð4Þ

Dz ¼ 1 þ c~ 2 ðg0 þ 1Þc~ 1 :

ð5Þ

Here the following correlation functions have been introduced: X s s 0 c1 ¼ 2/Ssf S c2 ¼ 2 /Ssf þ d S f þ d S, f þ d0 S, d

c~ 1;2 ¼ az c1;2 :

ð6Þ 0

The primed sum indicates that the term with d ¼ d is omitted in it. The structure factor Gk is defined by 1X Gk ¼ expðikdÞ, ð7Þ

g0

d

where g0 is the coordination number. Fourier transform //Skþ 9S k SSo can be written as X A 1 l,k : //Skþ 9S k SSo ¼ 2p l ¼ 1;2 oOl,k

ð8Þ

Here

Ol,k ¼ h þð1Þl o? k,

ð9Þ

2 2 ~ ðo? k Þ ¼ 2J g0 ð1Gk Þ½D? þ g0 b 1 ð1Gk Þ,

ð10Þ

D? ¼ 1 þ b~ 2 ðg0 þ1Þb~ 1 ,

ð11Þ

Al,k ¼ Z þð1Þl Jb1 g0

bl ¼

ð1Gk Þ

o?k

:

ð12Þ

ð1Þ

al þcl , 2

a~ þ c~ l b~ l ¼ l , 2

a1 ¼ 4Z2 þ u1 ,

l ¼ 1; 2,

a2 ¼ 4ðg0 1ÞZ2 þ u2 ,

f

where Sf is the spin-half operator at site f, d is a vector connecting nearest neighbors on a triangular lattice, J 4 0 is an exchange constant, h ¼ 2mB, m is the magnetic moment of a particle, and B is an external magnetic field. To calculate spin–spin correlators it is necessary to find two retarded commutator single-particle Green functions: //Szf 9Szf 0 SS, s 0 //Ssf 9S f SS ðs ¼ 7 Þ. We write down equations of motion for these two functions and make the decoupling of higher Green functions on the second step. In order to describe correctly the temperature region h oT o J at low magnetic fields h=J 5 1 we employ here a decoupling scheme, different from that used in our previous paper [38]. Namely s s s

Szi Szj Ssl ¼ ðZ2 þ a? /dSzi dSzj SÞSsl ,

The correlation functions entering Eqs. (10)–(12) are defined by

2. Method

f, d

s z s s z Ssi S j Sl ¼ az /Si Sj SSl ,

s s

s

s s

s

¼ a? ð/Sj Sl SSi þ/Si Sl SSj Þ,

a~ 1 ¼ 4Z2 þ a? u1 ,

a~ 2 ¼ 4ðg0 1ÞZ2 þ a? u2 ,

u1 ¼ 4/dSzf dSzf þ d S,

X0 u2 ¼ 4 /dSzf þ d dSzf þ d0 S:

ð13Þ

d

Due to the presence of the external magnetic field the quantity Z is nonzero at any finite temperature. An exact commutator Green function must not have any pole at o ¼ 0 [39,40], whereas the denominator of the approximate Green function (8) turns to zero at k ¼ k0 obeying the equation h ¼ o? k0 :

ð14Þ

To restore the correct analytical properties of the approximate function (8) the condition of vanishing the numerator of this function at k ¼ k0 must be satisfied [38]. From this condition we

I.I. Poltavsky et al. / Physica B 407 (2012) 3925–3932

get the following equation for Z: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 J @ 2h b~ 1 2 Z¼ D? þ 2 D? A, 2hK J

ð15Þ

where the parameter K ¼ b~ 1 =b1 has been introduced. Using the spectral relations [39] we have the set of equations for the correlation functions cl and ul 2Jc1 X u1 ¼ G g , N k k k 2 X G p , KN k k k

c1 ¼

2Jc1 X u2 ¼ ðg G2 1Þg k , N k 0 k c2 ¼

2 X ðg G2 1Þpk , KN k 0 k

ð16Þ

where N is the total number of sites pk ¼

Z~ sinhðbhÞJb~ 1 g0 ð1Gk Þsinhðbo?k Þ=o?k , coshðbhÞcoshðbo? Þ k

gk ¼

bozk g0 , ð1Gk Þcoth ozk 2





ð17Þ

Z~ ¼ KZ, b ¼ 1=T. The set (16) should be supplemented with equations for the vertex parameters az , a? and K. Two of them are chosen so as to satisfy the sum rules  2 s 4/ Szf S ¼ 1, 2/Ssf S f S ¼ 1 þ 2sZ, which can be rewritten in the form 2 X 1¼ p : KN k k

2Jc1 X g þ 4Z2 ¼ 1, N k k

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In the limiting case h¼0 the system (21) transforms into the set of equations [36] obtained within Kondo–Yamaji decoupling scheme.

3. Results and discussion In this section we calculate the temperature dependence of the magnetization MðTÞ ¼ 2ZM sat (where M sat ¼ Mð0Þ is the saturation magnetization) and use the obtained results to interpret the experimental data on magnetization for the second solid 3He monolayer on graphite and 3He monolayer on 4He-preplated graphite. Fig. 1 displays MðTÞ=M sat for 16  16 triangular lattice Heisenberg ferromagnet at different ratios h=J. It is seen that in the lowfield region the results of the present decoupling (2) demonstrate better agreement with QMC data [24] than those obtained within the scheme employed in our previous paper [38]. Another evidence in favor of the present theory is a very good agreement between QMC data [24] and our results for zero-field susceptibility of HFM on 32  32 triangular lattice at J ¼ 2:1 mK (see Fig. 2). The magnetization M(T) is principally sensitive to the cluster size N ¼ L  L, especially at small magnetic fields [38]. Fig. 3 demonstrates M=Msat as a function of temperature for h=J ¼ 0:1 and 0.001 at different N. For a given value of the magnetic field there exists the minimal cluster size N1 such that for N Z N1 the

ð18Þ

ð19Þ

The third equation follows from Eq. (13), and determination for K b~ K2 2Z~ 2 a? ¼ ~1 : b 1 K2Z~ 2

ð20Þ

Eqs. (15), (16), (19) and (20) represent a closed set of eight self-consistent equations for u1, u2, c1, c2, az , a? , K, and Z. This set can be reduced to three equations for b~ 1 , Dz , and D? 1¼

2Jc1 X g þ 4Z2 , N k k

1¼

" # 2b~ 1 2J X c1 1 ð1Gk Þg k , N k K

D? ¼ ð1a? Þð14Z2 Þ þ þ

a? 2az

Fig. 1. Temperature dependences of the magnetization for 16  16 triangular lattice HFM at h=J ¼ 0.858, 0.428, 0.0858, 0.0342, 0.0122, 5.52  103 , 8.58  104 (from top to bottom): present theory (solid), QMC [24] (symbols), and results of Ref. [38] (dashed).

ðDz 1Þ

J a? c1 X ð1Gk Þð1g0 Gk Þg k : N k

ð21Þ

All other quantities can be expressed through b~ 1 , Dz and D? according to Eqs. (15), (16), (19), and (20). For az we have

az ¼

1Dz : ðg0 þ1Þc1 c2

ð22Þ

Note that Eqs. (16) and (19) have no singularities due to condition (15). At arbitrary temperature the set of equations (21) can be solved numerically. The developed approach allows us to investigate the behavior of thermodynamic quantities at various relations between J and h as well as consider the infinite and finite-sized systems. In particular, in the case of small fields this method gives a proper description of HFM model in the temperature region h oT oJ, which is very important for interpretation of the experimental data on 3He (see Section 3).

Fig. 2. Susceptibility as a function of J=T for 32  32 triangular lattice HFM at B ¼0 and J ¼2.1 mK. Present theory (solid), QMC [24] ().

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Fig. 3. Magnetization vs temperature at h=J ¼ 0:1 (a), 0.001 (b) and different L (from bottom to top): 5, 7, 10, 25 (a), and 25, 50, 75, 150 (b). Present theory (solid), HTSE (dashed).

magnetization per spin is practically independent of N (thermodynamic limit). The smaller is the ratio h=J the larger is N 1 . Thus, at h=J ¼ 0:1 the quantity N1  400, whereas at h=J ¼ 0:001 the value N1  20; 000. The largest N in Fig. 3(a) and (b) is taken to exceed N1 , so that with further increase in N all dependences MðTÞ=Msat will coincide with the corresponding upper curves in these graphs. Fig. 3 also shows that the dependence of the magnetization on the cluster size is more pronounced in the low temperature region. With decrease in ratio h=J this region narrows. At h=J ¼ 0:1 it begins below T  J while at h=J ¼ 0:001 the magnetization becomes obviously size dependent only below T  0:5J. What is more, these are precisely the regions where MðTÞ=Msat changes rapidly with temperature. This fact is of principal importance when extracting the exchange constant J from experimental data. Usually, such extraction is made with the help of HTSE. Since the applicability of HTSE is restricted not only by large N but also by the condition T 4 J, at small fields it describes the magnetization only in a little-informative temperature region, where the value of MðTÞ=Msat is several orders less than unity. As a result, in the case of low enough fields the accuracy of extraction of the exchange constant J from the experimental data with the help of HTSE is substantially reduced. An important parameter of the system is the saturation magnetization M sat ¼ Mð0Þ which significantly depends on the coverage and layer structure. The quantity Msat is closely related to the number of active spins involved into ferromagnetic exchange [10,8]. The real measurements of the magnetization cannot be performed at arbitrary low temperatures, so that the direct determination of Msat is impossible. To extract this value it is necessary to extrapolate experimental dependences M(T) to the point T¼0. Up to now the approximate Kopietz formula [24] has been the only method for making such extrapolation. However, in view of the fact that the validity of this formula is restricted by too narrow region near T¼ 0, an accurate extraction of Msat with its help may break down, especially if available experimental data are beyond the temperature range where the analytical Kopietz expression can be applied with assurance. Furthermore, in the case of small clusters and low external magnetic fields the function M(T) rises sharply near zero temperature (see Figs. 1 and 3). This is another factor complicating the reliable extrapolation of M(T) data to T¼0. Now we proceed to analysis of experimental data known from the literature.

Fig. 4 displays the temperature dependences of the magnetization for the second 3He monolayer on an exfoliated graphite (Papyex) at an areal density 23.5 nm  2. The measurements were made on the same sample, so that the sets of the experimental data [12] differ only in the value of the magnetic field, whereas the spin exchange constant J and Msat are common for all the curves. Solid lines in Fig. 4 are the present theoretical results at fixed J¼1.725 mK, cluster size N¼1089, and different values of B taken in accordance with those used in the experiment. It is seen an excellent agreement between the present theory and experiment at all values of B in the whole temperature range where the measurements were made. Such agreement is a direct evidence in favor of applicability of HFM at this coverage. For comparison, the results for M(T) obtained in Ref. [12] by Kopietz formula at J¼1.9 mK and N¼1000 are also shown in Fig. 4. It is seen that this approximation is insufficient to describe the temperature region where the measurement was made. Besides, as our analyses show, at B ¼ 56 and 26 mT we deal with finite-sized system so that HTSE is also inappropriate for this case. The quantity N we find to fit the theory to experimental data has a meaning of an averaged cluster size. To be convinced that the extracted N is acceptable it is necessary to check up if the cluster of N spins can fit on an average graphite platelet. It turned out that all the experimental data we use here satisfy this condition. Fig. 5 illustrates the experimental [10] and theoretical temperature dependences of the magnetization at T o3 mK and fixed magnetic field B ¼14.3 mT for five different coverages. The exchange constant J and cluster size N chosen to obtain the best fit of the present theory to the experimental data are shown in Fig. 6. It is seen that the present theory successively interprets the experiment for all the coverages. As it was shown in Refs. [22,29] at low ferromagnetic densities such as r ¼ 20:5 and 21.5 nm  2 the MSE effects can contribute to the thermodynamics of solid 3 He monolayers. For these two coverages, however, there exist experimental measurements [10] at temperatures up to 5.5 mK, so that in this case we can check up the applicability of HFM in a wider temperature range. Fig. 7 displays the corresponding dependences M(T) completed by the high temperature data. To draw solid lines in Fig. 7 we take precisely the same values of J and N as in Fig. 5 at T o3 mK. From Fig. 7 it is seen that the present theoretical results and experimental data are in close agreement not only at low but also at high temperatures.

3.1. Pure 3He on graphite In this subsection we interpret a number of experimental data on magnetization for the second solid layer of pure 3He on graphite in the ferromagnetic regime at dense coverages.

Fig. 4. Temperature dependences of the magnetization for the second 3He monolayer on an exfoliated graphite (Papyex) at B¼ 113, 56, 26 mT (from top to bottom), r ¼ 23:5 nm2 . Experimental data [12] (symbols), present theory (solid), results [12] calculated with Kopietz formula (dashed).

I.I. Poltavsky et al. / Physica B 407 (2012) 3925–3932

3

Fig. 5. Temperature dependences of the magnetization for the second He monolayer on an exfoliated graphite at B ¼14.3 mT and (from bottom to top): r ¼ 20:5, 21.5, 22.7, 24.8, 25.9 nm  2. Experimental data [10] (symbols), present theory (solid).

Fig. 6. Exchange constant J (a) and cluster size N (b) as functions of coverage r that give the best fit of the present theory () to the experimental data [10]. The polynomial approximations (solid) and the results of Ref. [10] ().

Fig. 7. Temperature dependences of the magnetization for second 3He monolayer on an exfoliated graphite at B¼ 14.3 mT and (from bottom to top): r ¼ 20:5, 21.5 nm  2. Experimental data [10] (symbols) and present theory (solid).

This suggests that the behavior of M(T) at r ¼ 20:5 and 21.5 nm  2 can be treated with the help of HFM. In Ref. [29] it was shown that a pure Heisenberg behavior is observed at r Z22 nm2 when the exchange constants Jc and J w inferred respectively from HTSE heat capacity and susceptibility data coincide (see Fig. 4 in Ref. [29]). At r ¼ 21:5 and 20.5 nm  2 effect of MSE makes itself evident in slight difference between Jc and Jw [29]. Such

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Fig. 8. The coverage dependences of the saturation magnetization (m) and the relative number of clusters (). Symbols are the result of fit to the experimental data [10]. Solid lines are drawn by eye.

conclusions were made on the basis of HTSE calculations for an infinite MSE model in zero field. However, at coverages r ¼ 21:5 and 20.5 nm  2 we deal with finite-sized systems under external magnetic field. To estimate properly the multiple-spin contribution in this case it is necessary to have the solution of MSE model at these specified conditions. Now we consider the variation of the exchange constant J and cluster size N with coverage (see Fig. 6). In Fig. 6(a) the nonmonotonic behavior of JðrÞ is seen, when with increase in coverage the exchange constant first increases, goes through a maximum, and then decreases. Cluster size N is found to increase monotonically with density (see Fig. 6(b)). From our analysis it follows that at r \23 nm2 the thermodynamic limit is already reached, MðTÞ=N is independent of N, and the 3He second monolayer behaves like an infinite homogeneous ferromagnet, as it should be at such densities. At low coverages clusters are small, and strong spin fluctuations near their boundaries affect significantly the exchange reducing the average value of J [8,9]. As the cluster increases in size, the fraction of spins near its boundary decreases. This diminishes the boundary effects leading to a rise in the average exchange constant. When the thermodynamic limit is reached, the role of the boundaries becomes insignificant, and the value of J is completely determined by processes within the cluster. The saturation magnetization behaves with r in a similar way as NðrÞ (see Fig. 8). The grafoil surface used in the experiment [10] was composed of atomically smooth platelets with linear size ˚ From Fig. 6(b) one can see that at r ¼ 25:9 nm2 only about 100 A. one ferromagnetic island can be placed on a typical platelet. The ratio between number of clusters at given r and number of clusters at r ¼ 25:9 nm2 is shown in Fig. 8. It demonstrates the tendency of the cluster formation. At first with increasing coverage the number of clusters increases, then decreases showing that the clusters merge to become a homogeneous ferromagnetic system. The treatment of the experimental data in Fig. 5 with the Kopietz approximate formula leads to the following findings [10]. Entering this formula quantity N is usually identified with an effective cluster size. It behaves similarly to JðrÞ first increasing with r and then decreasing (open diamonds in Fig. 6). As noted in Ref. [10], this result contradicts to the experimentally observed continuous growth of the saturation magnetization with density, which implies an increase in the number of ferromagnetic spins, and, consequently, the growth of the cluster size. In our opinion, such inconsistency in the estimation of N is due to the fact that the Kopietz formula is appropriate at rather low temperatures. An

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Fig. 10. Temperature dependences of magnetization for ferromagnetic nanoclusters on an addition of 4He to the system. Experimental results [8] (symbols) and the present theory (solid curves).

Fig. 9. Temperature dependences of the magnetization. Experimental results (symbols) at (from top to bottom): B¼ 0.35 mT, r ¼ 24:2 nm2 (Ref. [7]); B¼ 6.44 mT, r ¼ 23:3 nm2 (Ref. [5]); B¼14.21 mT, r ¼ 23:3 nm2 (Ref. [6]); B¼ 30.5 mT, r ¼ 24:4 nm2 and B¼ 113 mT, r ¼ 38 nm2 (Ref. [8]). Present results (solid) are found at (from top to bottom): J¼1.65 mK, N¼10,000; J¼2.1 mK, N¼ 2500; J¼ 2.05 mK, N¼529; J¼2.15 mK, N¼ 144; J¼ 2.8 mK, N¼ 256.

attempt to apply this formula to the experimental data in as wide temperature range as possible results in an inaccurate estimate for N. Fig. 9 demonstrates the temperature dependences of the magnetization measured by different groups of experimentalists [5–8]. The results are obtained at various magnetic fields from high enough (113 mT) till very small (0.35 mT). In the case of two highest fields the measurements were made at dense coverages (large N) in a wide temperature range, including the region where the magnetization is close to saturation as well as temperatures T=J b1 where HTSE applies. This made it possible to extract the values of J and Msat combining the Kopietz formula and HTSE [8]. The quantities J and Msat found with the help of such procedure agree very well with the results of the present calculations. In particular, for B ¼113 mT we obtain exactly the same value of exchange constant J¼2.8 mK. Note, that the measurements at B ¼113 mT were taken at very high density r ¼ 38 nm2 when the third layer is considerably filled with 3He atoms. The applicability of the 2D Heisenberg model for this case is discussed in Ref. [8]. At B ¼ 14:21 and 6.44 mT there is a lack of low temperature experimental data, so that the Kopietz formula gives no way for deducing Msat with acceptable accuracy. As a result, the exchange constant cannot be extracted definitely from HTSE. The situation with M(T) at B¼0.35 mT is even more complicated. The corresponding measurements were made at h5 T o J, and neither HTSE nor Kopietz formula is applicable. On the contrary, the present theory is suitable to interpret these experiments. The best fit to the experimental data for B ¼0.35 mT gives N ¼ 104 . The obtained cluster size, even if very large, does not correspond to an infinite system due to extremely small magnetic field h=J  104 . 3.2.

3

He on 4He-preplated graphite

In this subsection we interpret the experimental results [8] obtained for solid 3He on 4He-preplated graphite. The important feature of the experiment [8] is the possibility to change the size and density of the ferromagnetic 3He regions by adding controlled amounts of 4He (for details see Refs. [8,11]). Fig. 10 displays the evolution of the magnetization curve of solid 3He monolayer as 4He is added to the system. At first, on addition of every successive portion of 4He the corresponding magnetization curve displaces and lies as a whole above the

Fig. 11. Ferromagnetic peak in the solid 3He on 4He-preplated graphite. Exchange constant J (a) and cluster size N (b) as functions of coverage that give the best fit of the present theory () to the experimental data [8]. Solid lines are guide by eye. Values of J and N from Ref. [8] are also displayed ().

previous one (see Fig. 10(a)). Such evolution continues until the coverage of 4He reaches the value of 9.15 ccSTP. With further increase in amount of 4He the magnetization decreases practically in the whole temperature range (see Fig. 10(b)). The dependences M(T) shown in Fig. 10 are related to coverages at which the HFM model definitely applies [29]. The solid lines are the results of the present calculations with J and N depicted in Fig. 11. Although the behavior of JðrÞ is qualitatively similar to that for pure 3He, the reasons for reducing the exchange constant at r 4 rm  24:3 nm2 are different. In the present case, both increase and decrease in J correlate with the change in cluster size, contrary to the pure 3He where after the ferromagnetic peak the clusters continue to grow. As follows from Fig. 11(b), the size of 3He clusters varies nonmonotonically with the addition of 4He atoms. First, NðrÞ grows due to increase in the second layer density. The higher is r the more 3He atoms are involved into ferromagnetic exchange. When the amount of 4He atoms reaches the value  9 ccSTP the function NðrÞ goes through the maximum and then an average cluster size decreases. Since the processes before and after the ferromagnetic peak occur at somewhat different pressures, the dependences NðrÞ and JðrÞ are not symmetric with respect to the maxima. The coverage dependences of the saturation magnetization and the ratio between number of clusters at given r and number of clusters at r ¼ 26:3 nm2 are shown in Fig. 12. At r t rm the average number of spins per cluster grows with coverage, the clusters merge, and their number Ncl reduces. With further increase in r the number of clusters increases while their size decreases. The function M sat ðrÞ is nonmonotonic and reaches the maximal value at the coverage that exceeds rm . This is because the saturation magnetization is proportional not only to the

I.I. Poltavsky et al. / Physica B 407 (2012) 3925–3932

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Fig. 14. Temperature dependences of the magnetization calculated by exact diagonalization at B¼ 30.5 mT, J¼ 2 mK, N ¼ 12, free boundary conditions, and different shapes 3  4 (upper) and 2  6 (lower).

Fig. 12. The coverage dependences of the saturation magnetization (m) and the relative number of clusters (). Symbols are the result of fitting the present theory to the experimental data [8]. Solid lines are drawn by eye.

Fig. 14 demonstrates the temperature dependences of the magnetization for two nanoclusters equal in size N ¼ 12 but distinct in shape. The number of interspin bonds in the 3  4 cluster is greater only by two than that in the 2  6 cluster. Although small enough, such a difference leads to a noticeable distinction in the behavior of the corresponding magnetizations. However, varying the exchange J it is possible to set off the effect of the cluster shape and to obtain close magnetization curves for both nanoclusters. From the above it follows that for small clusters unambiguous extraction of the exchange constant can be done only with additional information on the cluster shape and boundary conditions.

4. Conclusions

Fig. 13. Temperature dependences of the magnetization at B¼ 30.5 mT. Experimental data [11] (m), the results of exact diagonalization at N ¼ 15: for free boundary (solid lines) at J¼ 1 mK (lower) and J¼ 1.6 mK (upper), and for periodic boundary condition at J¼ 1 mK (J).

average cluster size but also to the number of clusters in the system. A peculiar feature of the system under consideration is that it consists of very small clusters. For instance, our calculations show that the lowest curve in Fig. 10(b) represents the magnetization for a nanocluster of 16 particles. In Ref. [11] the corresponding experimental data were interpreted using ED and QMC. It was obtained that the average cluster size is 16 73 and the exchange J¼2 mK. While the cluster sizes obtained both by the present analytical method and numerical simulations [11] are the same, the exchange constants differ practically twofold. The origin of this discrepancy consists in a principal dependence of the magnetization for small clusters on the cluster shape and boundary conditions. To illustrate these facts we made the corresponding ED calculations for the cluster of N ¼15 spins. Fig. 13 shows the effect of boundary conditions on the shape of the magnetization curve. The lower solid line and the data presented by open circles are calculated at J¼1 mK and differ only in boundary conditions (free and periodic, respectively). The periodic condition means the presence of additional exchange for the boundary spins which leads to increase in the magnetization as compared to the case of free boundary. However, it is possible to get close results at both boundary conditions by variation in exchange J. Thus, the upper solid curve calculated for the free boundary at J¼1.6 mK is indistinguishable by eye from the corresponding curve for the same cluster at periodic boundary conditions and J¼1 mK. These both results describe well the experimental observations.

In the present paper we develop an analytical approach for the 2D ferromagnetic Heisenberg model in an external magnetic field that describes properly the thermodynamics of infinite and finitesized spin systems at arbitrary temperatures and fields. Compared to other theories, the proposed method substantially improves the description at low magnetic fields h=J 5 1 in the intermediate temperature region h oT oJ. It is precisely these conditions that are used in experiments on 2D solid 3He monolayers. Owing to the developed theory we give a consistent interpretation to a great number of experimental data on magnetization of 3He monolayer on graphite and 3He on 4He-preplated graphite in the ferromagnetic regime. Fitting the present theory to the experimental data we extract the exchange constant J, saturation magnetization Msat, and average cluster size N at given h. Below the thermodynamic limit the shape of magnetization curves MðTÞ=Msat at h=J 5 1 is found to be extremely sensitive to the cluster size just in the temperature region h o T oJ. Due to the proper description of the magnetization for finite-sized systems at intermediate temperatures, the proposed theory provides an unambiguous extraction of N from the experimental data. For pure 3He on graphite we succeed in interpretation of experimental data for M(T) measured on the same sample at the fixed coverage for different values of the magnetic field. Under this condition J and N are the same for all dependences M(T). Once the values of J and N are identified, each theoretical curve M(T) at given B falls precisely on the corresponding experimental points. An excellent agreement between the theory and experiment is a direct evidence in favor of applicability of the Heisenberg model. We also interpret the behavior of experimentally obtained magnetization curves M(T) at fixed B and different coverages. Although the Heisenberg model is found to be appropriate at r Z 22:0 nm2 [29], we show that it provides a reasonable description for M(T) at r ¼ 20:5 and 21 nm  2. On the basis of these results, the coverage dependences of the exchange constant, saturation magnetization, and average cluster size are obtained

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and analyzed. The exchange constant displays nonmonotonic behavior with increasing coverage. It first increases, goes through a maximum, and then decreases. The cluster size N as well as Msat continuously grow with total coverage. Using the present method we interpret the evolution of magnetization curves for solid 3He on 4He-preplated graphite at fixed B as portions of 4He are added to the system. We find the coverage dependences of J, N, and Msat. In this case both JðrÞ and NðrÞ behave nonmonotonically. The function NðrÞ reaches a peak just at the coverage corresponding to the maximum of JðrÞ and then sharply decreases. For small clusters (nanoclusters) we show by exact diagonalization method that the magnetization strictly depends on the shape of the cluster and the boundary conditions. Thus, we have shown that the Heisenberg model is appropriate for description of all the experimental data considered in this paper. However, to make a final conclusion about areas of application of HFM model to 2D solid 3He it is desirable to have experimental data for the magnetic susceptibility and heat capacity of the second 3He monolayer obtained on the same sample at non-zero field and the theoretical calculations for MSE model in an external magnetic field within a whole temperature range. References [1] [2] [3] [4] [5] [6] [7]

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