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This paper attempts to develop a unified theoretical model to describe the size dependence of the Curie temperature for materials of different sizes. To this end, the sizedependence model of glass transition temperature was expanded into a sizedependence model of the Curie temperature Tc of uniform lowdimensional ferromagnetic materials, based on energy theory and thermodynamics. The proposed model focuses on the critical size and the material shape, and has no adjustable free parameter. The model was used to predict the effect of material size on the Curie temperature, revealing that the Tc value decreases with the material size. This result agrees well with the experimental results and the prediction by other theoretical models. The research findings make up for the gap in the previous research on the Curie temperature.
low dimension, crystal, curie temperature
With reduced symmetry, ferromagnetic nanometals have different properties from bulk metals. The size of these materials has direct impacts on local surface properties like valence band structure and membrane shape. In return, these properties can serve as sensitive functions of material size and growth conditions [1].
In terms of application, a nanocrystal can be divided into two categories, namely, a thin membrane less than 10nm in thickness [23] and a bulk material consisting of nanoparticles or nanocrystals [4]. In recent years, much research has been done on the magnetic properties of membranes, especially the temperature of magnetic phase transition, i.e. the Curie temperature [23], revealing that the Curie temperature changes with the material size. On this basis, some theoretical models have been developed to describe the variation of Curie temperature with sizes [13].
One of these models is grounded on magnetic pole coupling. For a thick membrane, the Curie temperature satisfies:
T_{c}/T_{c0 }$\approx $ 1[(n_{0}+1)/2n]^{λ} (1)
where Tc0 is the Curie temperature of the bulk crystal; λ∈[1, 1.59] [2] is a constant dependent on the magnetic pole coupling; n=S/h is the number of deposited membrane layers (MLs), with S being the membrane thickness and h being the layer spacing (i.e. atomic diameter). If the membrane thickness becomes sufficiently small n0, there will be a linear relationship between the Curie temperature and the material size:
$\frac{T_c}{T_{c0}}=\frac{n1}{2n_0}$ (2)
The above model predicts that Curie temperature decreases with the membrane thickness, which has been validated by experimental results.
Another model, created on the sequence, length and strength of keys [5], can also accurately depict the Curie transition of ferromagnetic membranes.
Meanwhile, the unique physicalchemical properties of ferromagnetic particles have also attracted attention in basic and applied research. For these particles, the Curie transition is essentially the shift of magnetic domains from disordered alignment to ordered alignment. The shift takes place right after the decline in particle size D, a.k.a. particle/grain diameter. To our knowledge, there is no theoretical model on the size dependence of the Curie temperature of ferromagnetic particles, which differs from that of membranes [2]. Thus, it is necessary to develop a unified theoretical model for particles with various sizes d.
The previous studies have shown the dependence of Curie transition on the material shape and size [2]. As a result, the physical nature of the transition should be fully considered in both theory and application.
In addition, the Curie transition is a secondary phase transition. Its features can be derived from those of other secondary phase transitions like glass transition. For glass transition, the sizedependence of the transition temperature T_{g} has been obtained for various lowdimensional materials by extending the sizedependence model of the melting temperature [67].
In this paper, the sizedependence model of glass transition temperature is expanded to examine the size dependence of the Curie temperature for magnetic materials in different sizes. Special attention was paid to determining the critical size D_{0}, i.e. the size of a singledomain particle/domain. The determination requires different methods for different material sizes. Several experiments were carried out on Fe, Co and Ni membranes as well as Ni nanoparticles and Ni nanowires. The prediction of our model was found to be consistent with the experimental results.
Replacing the glass transition temperature T_{g} and bulk material transition temperature T_{g0} with the Curie temperature T_{c} and bulk material Curie temperature T_{c0} [67], the glass transition temperature T_{g} model can be modified as:
$\frac{{{T}_{c}}}{{{T}_{c0}}}=\exp \left( \frac{2\Delta {{C}_{p}}}{3R}\frac{1}{(\frac{D}{{{D}_{0}}1})} \right)$(3)
where, ΔC_{p} is the specific heat difference of the ferromagnetic body at the Curie temperature T_{c}_{0}; R is the ideal gas constant; D_{0} is the critical size of the material that induces the Curie transition.
The critical size D_{0} is usually defined as the diameter of all atoms on the surface of a lowdimensional material. If the dimension k of a spherical particle is zero, then the critical size D_{0} equals six times the atomic diameter h; if k=1 (i.e. the material is a nanowire), then the critical size equals four times the atomic diameter; if k=2 (i.e. the material is a membrane), then the critical size equals twice the atomic diameter.
The material size D directly bears on the magnetic transition of lowdimensional magnetic materials. The magnetic transition takes place when the temperature reaches a critical value T_{c}, turning the ferromagnetic body into paramagnetic body. In other words, the magnetic domains change from ordered alignment to disordered alignment. The magnetic transition is reversible. If the temperature decreases, the paramagnetic body will change back into ferromagnetic body, and the magnetic domains will become more orderly. Here, 2D_{0} is defined as the minimum material size for magnetic domains to remain orderly.
With low magnetic permeability and high coercive force, singledomain particles have a great influence on material properties and critical size estimation. These particles can only be magnetized rotationally, due to the absence of domain walls, and cannot be easily magnetized or demagnetized without external field or external force.
The critical size is the division point between the singledomain and other domain structures. Thus, the energy of the critical size of the singledomain structure is comparable to that of the simplest adjacent multidomain structure. The two structures will have equal energy if both are at the critical size. Thus, the critical size of a single domain can be identified using the comparability [8].
The domains in ultrafine particles are spherical. For such a spherical domain, the total energy can be calculated as E_{T}=E_{w}+E_{H}+E_{d}, where E_{w} is the domain wall energy, E_{H} is the magnetostatic energy, and E_{H} is the demagnetizing energy E_{d} [9]. Without external magnetic field, there is naturally no magnetostatic energy. The demagnetizing energy can be neglected because the domain surface only has a weak magnetic pole. Hence, the total energy of a spherical domain without external magnetic field is equivalent to the domain wall energy [9]:
$E_T=E_{\omega}=S\sigma /2$ (4)
where, S=πD^{2} and σ are the area and the energy density of the domain wall, respectively; D is the diameter of the domain. The energy density varies from domain wall to domain wall.
For the 90° domain wall in cubic crystals, σ equals $\pi \sqrt{A_1K_1}/2$; for the 180° domain wall in uniaxial crystals, σ equals $4\sqrt{A_1K_1}$ [10]. Here, A_{1} is the exchange integral constant and K_{1} is the magnetic anisotropy constant.
For singledomain crystals, the magnetocrystalline anisotropy energy is minimized when the magnetic moments are arranged in parallel along the easy magnetization axis. In the absence of external magnetic field and internal stress, neither the magnetostatic energy nor magnetoelastic energy of the external field needs to be considered. Coupled with the lack of energy exchange, it is only necessary to consider the demagnetizing energy. Hence, the total energy of a singledomain crystal can be expressed as [10]:
$E_T=E_d=\mu_0 VNM_s^2/2$ (5)
where, μ_{0} is the magnetic constant or a vacuum permeability; M_{s} is the saturation magnetization intensity; V=πD^{3}/6 is the volume of the domain; N is the demagnetization factor. Here, N=N_{a}α_{a}^{2}+N_{b}α_{b}^{2}N_{c}α_{c}^{2}, with α_{i} (i=a, b, c, the three major axes of the crystal) being the direction cosine of the saturation magnetization intensity M_{s}, and N_{a}, N_{b} and N_{c} being the demagnetization factors along the three axes (N_{a}+N_{b}+N_{c} =1), respectively. For spherical particles, the major axes are equal in length (a=b c) such that N_{a}=N_{b}=N_{c}. Thus, the demagnetization factor N of a spherical particle equals 1/3 [10]. According to formulas (2) and (3), the critical size D_{0} can be derived as:
$D_0=\frac{9\sigma}{\mu_0M_S^2}$ (6)
The size dependence of the Curie temperature for magnetic nanoparticles can be obtained from formulas (3) and (6). It can be seen from the formulas that the Curie temperature increases exponentially with the reduction in particle size.
For wires and membranes, the ferromagnetic coupling is sufficiently strong to make the magnetic momentum parallel to the membrane direction [1], when the wire/membrane thickness n (the number of atomic layers) is greater than 20~30. Therefore, the size dependence of the Curie temperature T_{c} for the wire/membrane can be predicted directly by formulas (3) and (6).
If n is smaller than 20~30, most ferromagnetic membranes will change from 2dimension to quasi1dimension [11]. In this case, the membranes exist as an islandlike deposition on the substrate, and the critical size D_{0} should be recalculated. For these membranes, the domain size is no longer the membrane thickness, but the diameter of islandlike membrane.
By firstorder approximation, the shape of islandlike membrane can be viewed as a flat sphere or a spherical crown. Then, we have S=πD't, V=π(D')^{2}t/4, with D' and t being the diameter and thickness of the flat spherical domain, respectively. As discussed above, D'_{0} equals 2σ/(μ_{0}NM_{s}^{2}) if E_{ω}=E_{d}. The islandlike membrane can be considered as a quasi1D system, because it is an intermediate form between particle (0D) and thin membrane (2D) [94]. Then, the demagnetization factor N of the islandlike membrane equals the arithmetic mean (2/3) of the demagnetization factor (N=1/3) of the particle and the demagnetization factor (N=1) of the membrane.
The relationship between D' and D can be determined according to the Gibbs free energy difference ΔG between the islandlike membrane and the normal membrane [94].
Here, a square membrane (side length: L; thickness: D; volume: V=L^{2}D) is cited as an example for the calculation of ΔG. Under a certain temperature T, the membrane will transform into a flat spherical particle of the same volume. In this case, the Gibbs free energy difference ΔG can be calculated as VΔP_{0}+(γ_{2}S_{2} γ_{1}S_{1}), where, ΔP_{0}=P_{1}  P_{0} is the pressure difference between the islandlike membrane and the normal membrane; S_{2} = π(D')^{2}/4+π(D')^{2}/4+πD't is the sum of the surface area of the islandlike membrane and the area of the interface between the membrane surface and the substrate; S_{1} = L^{2}+L^{2}+4LD is the sum of the surface area and the interface area before the formation of the islandlike membrane; γ_{1} and γ_{2} are the surface energies (interface energies) before and after the transition, respectively. Note that P_{1} refers to the pressure difference between the inside and outside of the membrane resulted from the surface curvature of the islandlike membrane. According to the Gibbs–Thomson equation, P_{1 }equals 2f/D', with f being the surface stress of the crystal [12]. For the normal membrane, P_{0 }equals zero as the surface is flat with no curvature. In addition, the first and second terms in S_{2} represent the surface area of the upper portion and the interface area between the lower portion and the substrate, respectively.
Thus, it can be derived that ΔG=V(2f/D'+γ/t+γ'/t+4γ/D'γ/Dγ'/D4γ/L), where γ is the surface energy of the crystal; γ' is the solid interface energy between the membrane and the substrate. Considering the material difference across the interface between the membrane and the substrate, the value of γ' can be computed by calculating the solid interface energy between the same materials and taking the arithmetic mean as the solid interface energy between two different materials. The values of f and γ_{ss} can be calculated as [13]:
$f=\pm \frac{7}{2\left(\frac{T_m}{T+6}\right)}\sqrt{3h^2S_{vib}H_m/\left(\kappa RV_m\right)} $ (7)
$\gamma=\frac{196hT^2H_mS_{vib}}{3R{(T_m+6T)}^2V_m}$ (8)
where T_{m} is the melting point of the material; h is the atomic diameter; H_{m} is the melting enthalpy; S_{vib} is the vibrational entropy in the melting entropy S_{m} (S_{vib}$\approx $ S_{m} for metals [63]); R is the ideal gas constant; V_{m} is the molar volume; k=1/B is the compression factor, with B being the bulk modulus.
Once the membrane thickness is reduced to a certain extent, the uniform membrane will spontaneously transform into the islandlike membrane if the Gibbs free energy difference ΔG_{0} between the two membranes is below zero. Mathematically, the spontaneous transition will take place if ΔG £ 0. Since V=L^{2}D=π(D')^{2}t/4 and L=D' at the critical point, it is possible to deduce that D = πt/4. Then, the transition condition can be written as D'/D³8f/[(4π)(γ+γ')]. Assuming that D'/D=C, we have:
$C=8f[(4\pi)/(\gamma+{\gamma}')]$ (9)
Substituting D'/D=C into D'_{0}=σ/(μ_{0}NM_{s}^{2}), we have:
$D_0=\frac{3\sigma}{\mu_0CM_s^2}$ (10)
The T_{c} curve of Ni particles and wires is illustrated in Figure 1, where the solid line depicts the T_{c} curve of Ni particles and wires obtained by formulas (3) and (6), the solid dot stands for the experimental results of Ni particles [4], and the hollow dots and plus sign represent the experimental results of Ni wires [14, 15].
Considering the 90° domain walls of Ni particles and wires, the value of $\sigma$ must be $\pi \sqrt{A_1K_1}/2$. For large nanowires, the size dependence of the Curie temperature can also be theoretically obtained from formula (3), after redefining D as the diameter of the nanowire [14]. As shown in Figure 1, both the theoretical and experimental results agree that T_{c} decreases with the particle size D.
Figure 1. The T_{c} curve of Ni particles and wires
Figure 2. The T_{c} curves of Fe, Co and Ni membranes
The T_{c} curves of Fe, Co and Ni membranes deposited on different substrates with coherent interfaces are shown in Figure 2, where the solid lines stand for the results predicted by formulas (3) and (10); the dash lines reflect the results predicted by formulas (1) and (2); the solid boxes represent the experimental results for Fe [23]; the solid dots, hollow dots and plus sign jointly describe the experimental results for Co [24]; the solid up arrows, hollow boxes, solid left arrows, solid down arrows and solid boxes joints represent the experimental results for Ni [2527].
Considering the 180° domain walls of the membranes, the value of $\sigma$ must be $4\sqrt{A_1K_1}$. As shown in Figure 2, the theoretical predictions on the T_{c} of Fe, Co and Ni membranes agree well with the experimental results, indicating that the material size changed from 2dimension to quasi1dimension with the reduction in membrane thickness. Since the magnetic order relies on material size, it is concluded that the dimensional transition has a certain impact on the ordering of magnetic domains. In addition, the T_{c} of membranes declined much faster than that of large particles and wires. This is because the islandlike membrane has a much smaller D_{0} than particles.
Compared with the model based on spin interaction (formulas (1) and (2)), formula (3) can predict the theoretical value of T_{c} excellently using the unified T_{c} equation only. Moreover, that formula (3) has no free parameter, while both n_{0} and l are fitting parameters in formulas (1) and (2).
Our model also discloses the relationship between membrane shape and membrane thickness from the angle of thermodynamics. The low surfacetovolume ratio was found to be the primary cause for the membrane to change from a uniform 2dimension material to an islandlike membrane. The model can also reveal the size dependence of ferroelectric transitions, after redefining 2D_{0} as the transition limit [28].
Table 1. Parameters and data in Figures 1 and 2

Fe 
Co 
Ni 
T_{c0 }(K) [18] 
1042 
1395 
631 
A_{1 }(10^{11}J/m) [18] 
1.21 
1.5 
0.67 
K_{1 }(J/m^{3}) [18] 
4.8×10^{4} 
4.3×10^{4} 
4.5×10^{3} 
M_{s} (A/m) [16] 
1.712×10^{6} 
1.422×10^{6} 
4.85×10^{5} 
ΔC_{p }(Jgatom^{1} K^{1}) [19] 
15.24 
10.9 
5.06 
h (nm) [20] 
0.344 
0.334 
0.324 
V_{m} (cm^{3}mol^{1}) [20] 
7.1 
6.7 
6.59 
H_{m} (KJ mol^{1}) [20] 
13.80 
16.19 
17.47 
T_{m} (K) [20] 
1809 
1768 
1726 
S_{m} (J mol^{1} K^{1}) 
7.63 
9.16 
10.12 
ĸ (10^{12}Pa^{1}) [21] 
5.889 
5.510 
5.640 
f (J/m^{2}) 
3.69 
4.56 
3.984 
γ (J/m^{2}) [22] 
2.43 
2.78 
2.01 
γ_{ss} (J/m^{2}) 
0.275 
0.407 
0.491 
γ' (J/m^{2}) 
0.249 
0.364 
0.406 
C 
12.80 
13.52 
15.39 
Considering the critical size and material shape, this paper puts forward a sizedependence model of the Curie temperature T_{c} of uniform lowdimensional ferromagnetic materials, based on energy theory and thermodynamics. The proposed model, with no adjustable free parameter, was used to predict the effect of material size on the Curie temperature, revealing that the T_{c} value decreases with the material size. This result agrees well with the experimental results and the prediction by other theoretical models.
This paper is made possible thanks to the support from the National Natural Science Foundation of China (Grant No.: 51878316).
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