1 Introduction

Magnetocaloric effect (MCE) arises from the entropy change of magnetocaloric materials (MCMs) due to magnetic field variation [1,2,3,4]. Because MCMs are generally solid and compact, such that magnetic refrigeration technology based on MCMs presents no Global Warming Potential (GWP) or Ozone Depletion Potential (ODP). For this reason, MCMs have been extensively explored for its potential application in the room temperature (RT) magnetic refrigeration [5, 6].

Magnetocaloric refrigeration material gadolinium (Gd) [7] as the working substance was used in RT magnetic refrigeration experimental machine in 1976. Afterward, a variety of magnetocaloric materials with a large magnetocaloric effect have been found and explored, such as Gd5Si2Ge2 [7], MnFeP0.45As0.5 [8], LaFe11.4Si1.6 [9], and La1-xCaxMnO3 (x = 0.2, 0.33) [10]. However, only under 5 T external magnetic field change can these materials be magnetized saturatedly, stimulating a lattice structural transition and resulting in a giant magnetocaloric effect (GMCE), named first-order transition (FOT) material [11]. However, the temperature region where the large magnetic entropy changes occurs in a narrow temperature range. On the other hand, some magnetocaloric materials only exhibit ferromagnetic-paramagnetic phase transition [12, 13], named second-order transition (SOT) material. Compared with the FOT materials, the entropy change of SOT materials is lower, but it does not possess thermal hysteresis and magnetic hysteresis [14]. Therefore, there were some scholars and engineers paying more attention to searching for SOT materials with large magnetic entropy change [15, 16].

Among the SOT materials, Gd, whose Curie temperature is 293 K, was considered to be an ideal working substance for RT magnetic refrigerant and has been efficiently utilized in current prototypes [17]. Furthermore, Gd-based rare earth alloys such as gadolinium-terbium (Gd-Tb), gadolinium-holmium (Gd-Ho), gadolinium-dysprosium (Gd-Dy), and gadolinium-erbium (Gd-Er) can be used as the RT magnetic refrigerant due to enhanced MCE and the adjustable Curie temperature around RT [18, 19]. Balli et al. [18] and Hou et al. [17] separately found that the doping of Tb and Dy into Gd can adjust the Curie temperature of Gd, where Gd0.7Tb0.3 and Gd0.73Dy0.27 can reach a maximal entropy change of 10.2 J/(kgK) under 5 T field variation and adiabatic temperature change of 3.1 K at 260 K under 1.2 T field variation. However, these Gd-based rare earth materials generally exhibit a caret-like entropy curve as temperature, and it is conflicting with that a perfect Ericsson refrigeration cycle requires a constant isothermal entropy change of the working substance over operating temperature region [14]. Thus, in order to obtain a relatively gentle entropy change curve of magnetocaloric material, some researchers focused their attentions on the study of the magnetocaloric effect of composite. For example, Bao et al. [20] found that the composite material composed with Gd50Al30Co20 and Gd55Al20Co25 can obtain nearly a constant entropy change 8.69 J/(kgK) under 5 T field variation. Ma et al. [21] found that a composite, composed of Gd50Co48Fe2 and Gd50Co48Mn2 with a mass ratio of 7:3, could reach 4.32 J/(kgK) isothermal entropy change between 255 K and 275 K under 5 T field variation. Zhong et al. [22] found that the multilayer composite composed of Gd65Mn25Si10 and Gd (mass ratio of 7:3) exhibits a large MCE with enhanced δTFWHM 212 K of the isothermal magnetic entropy change curve as well as large RC of 724 J/kg under 5 T field variation. Xu et al. [23] investigated that the composite, composed of Gd, Gd0.91Ho0.09 and Gd0.8Ho0.2 with a mass ratio of 0.59:0.17:0.24, can obtain a nearly constant magnetic entropy change of 4.09 J/(kgK) and found that the regenerative Ericsson refrigeration cycle using the composite material composed with Gd-Ho alloy exhibits a better performance than that of the component materials.

The composite material possesses a gentler isothermal entropy change curve than the component materials. But we also need some other parameters to evaluate the MCE and refrigeration performance of composite material. Therefore, the RC and RCP are introduced to compare the magnetic material, including the composite. In this paper, we analyzed the magnetocaloric characteristics including the maximal entropy change, full width at half maximum of the isothermal entropy change curve (δTFWHM ), RCP and RC of Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20), as well as the composite composed with Gd and Gd0.80Ho0.20 alloy. The results obtained can provide some references for the selection and parametric design of RT magnetic refrigeration materials.

2 Thermodynamic Analysis of Materials

2.1 Thermodynamic Analysis of Single Element Magnetocaloric Material

First of all, considering a single element ferromagnetic material, its magnetic entropy S(H, T) can be written as [24]

$$ S\left(H,T\right)=R\left\{\ln \left[\frac{\sinh \left[y\left(2J+1\right)/2J\right]}{\sinh \left(y/2J\right)}\right]-{yB}_J(y)\right\} $$
(1)

where R and J are universal gas constant and angular quantum number, respectively, and BJ(y) = coth[y(2J + 1)/2J](2J + 1)/2J − (1/2J) coth(y/2J) is the Brillouin function. Additionally, y can be obtained from the state equation of ferromagnetic material, i.e.,

$$ M= NgJ{\mu}_B{B}_J(y) $$
(2)
$$ y=g{\mu}_B{\mu}_0J\left(H+\lambda M\right)/ kT $$
(3)

where N is the number of magneton per unit volume, g is the Lander factor, μB is the Bohr magneton, μ0 is the vacuum permeability, H (B = μ0H) is the external magnetic field, λ is the molecular field coefficient, k is the Boltzmann constant, and T is the absolute temperature. For the given external magnetic field and temperature, the magnetization M and y can be solved out through Eqs. (2) and (3). Subsequently, according to Eq. (1), the magnetic entropy can be calculated.

2.2 De Gennes Factor Model Applying to Binary Alloys

Note that Eqs. (1), (2), and (3) can be used to calculate the magnetic entropy of single element ferromagnetic material. For binary alloys Gd1-xHox, we may employ the de Gennes factor model method. The expressions of the de Gennes factors and the effective magnetic moment for the alloy Gd1-xHox are given by [25].

$$ \overline{G}={\left(\overline{g}-1\right)}^2\overline{J}\left(\overline{J}+1\right)=\left(1-x\right){G}_{Gd}+{xG}_{Ho} $$
(4)
$$ {\overline{\mu}}^2={\overline{g}}^2\overline{J}\left(\overline{J}+1\right){\mu}_B^2=\left(1-x\right){\mu}_{Gd}^2+x{\mu}_{Ho}^2 $$
(5)

where \( \overline{g} \) is the effective Lander factor; \( \overline{J} \) the effective angular moment number; x is the molar fraction of Ho in the alloy Gd1-xHox; GGd and GHo are the de Gennes factors; and μGd and μHo are the effective magnetic moments of Gd and Ho material, respectively, and GGd/Ho and μGd/Ho can be calculated through Eqs. (6) and (7):

$$ {G}_{Gd/ Ho}={\left({g}_{Gd/ Ho}-1\right)}^2{J}_{Gd/ Ho}\left({J}_{Gd/ Ho}+1\right) $$
(6)
$$ {\mu}_{Gd/ Ho}={g}_{Gd/ Ho}\sqrt{J_{Gd/ Ho}\left({J}_{Gd/ Ho}+1\right)}{\mu}_B $$
(7)

where gGd/Ho and JGd/Ho are the Lander factor and angular quantum number of Gd/Ho. Based on Eqs. (4), (5), (6), and (7) and the relevant data of Gd and Ho listed in Table 1, \( \overline{g} \), \( \overline{J} \), and \( \overline{G} \) as well as \( \overline{\mu} \) can be obtained. On the other hand, the Curie temperature Tc of Gd1-xHox alloy can be calculated from the following equation [27]:

$$ {T}_c=46.63{\overline{G}}^{2/3} $$
(8)
Table 1 Relevant parameter values of Gd and Ho [26]
Full size table

3 Relative Cooling Power and Refrigeration Capacity

Using the Lander factor and angular quantum number of Gd1-xHox alloy, the magnetic entropy of Gd1-xHox can be calculated by using Eqs. (1), (2), and (3). Furthermore, for a given magnetic field or field variation, the isothermal magnetic entropy change curve of Gd1-xHox as well as the δTFWHM can be obtained. In order to compare the refrigeration performance of the magnetocaloric materials, there are two parameters; the relative cooling power (RCP) and refrigeration capacity (RC) [28,29,30,31] should be adopted. The RCP is defined as the product of (–ΔSmax) and δTFWHM of the isothermal magnetic entropy change curve, that is,

$$ RCP=-\Delta {S}_{\mathrm{max}}\times \delta {T}_{FWHM} $$
(9)

while RC is determined as

$$ RC={\int}_{T_{r2}}^{T_{r1}}-\Delta S\left(\Delta B,T\right) dT $$
(10)

where Tr1 and Tr2 are the temperatures of the cold and the hot end at half height of maximal entropy change curve respectively.

4 Results and Discussion

4.1 The MCE Characteristics of Gd1-xHox Alloys

Magnetic materials, due to their space lattice and electron motion differences, may be classified as diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, and ferrimagnetic ones. For room temperature magnetic refrigeration, MCE properties of ferromagnetic and paramagnetic materials are often utilized. It is well known that Gd material shows a paramagnetic to ferromagnetic transition when the temperature decreases or an external magnetic field is applied. Similarly, the Gd1-xHox also experiences the same magnetic phase transition when Ho concentration is lower than 0.25 [32]. However, though the doping of Ho does not change the magnetic material phase transition category fundamentally, it can influence the magnetic material properties such as angular moment number, Lander factor, consequently the magnetocaloric properties such as Curie temperature, magnetic entropy change, and so on. Based on the relevant data of Gd and Ho in Table 1, the effective Lander factor, effective angular moment number, and Curie temperature of Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20) can be obtained by using de Gennes factor model method. Subsequently, the theoretical isothermal entropy change of Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20) curves as temperature under 0 to 2 T field variation are obtained, as shown in Fig. 1a. And Fig. 1b shows the comparison between the experimental data [33] and the theoretical entropy change of Gd0.80Ho0.20. Figure 1b indicates that the experimental entropy change of Gd0.80Ho0.20 under 1 T field variation fits well enough with the theoretical ones, which is good support of the de Gennes method used in this paper. Moreover, it also can be clearly seen that the isothermal entropy change curves of Gd1-xHox alloys (x = 0, 0.05, 0.10, 0.15, 0.20) versus temperature all exhibit a caret-like shape curve. Furthermore, as Ho concentration increases, the Curie temperatures of Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20) are 293 K, 286 K, 278.9 K, 271.7 K, and 264.4 K, and maximal isothermal entropy variations are 5.63 J/(kgK), 5.78 J/(kgK), 5.94 J/(kgK), 6.11 J/(kgK), and 6.29 J/(kgK) under 2 T field variation, respectively. Additionally, it can be seen in Table 2 that both RCP and RC of Gd1-xHox increase with Ho concentration, where the δTFWHM approximately keeps constant, about 47 K, and Tr1 and Tr2 shift to low temperature region.

Fig. 1
figure 1

a The theoretical isothermal entropy change versus temperature curves of Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20) under applied magnetic field variation from 0 to 2 T. b The experimental [33] and theoretical entropy change of Gd0.80Ho0.20 under 1 T field change

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Table 2 The MCE and refrigeration characteristics of Gd and Gd1-xHox alloys under applied magnetic field variation from 0 to 2 T
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4.2 The MCE and Refrigeration Characteristics of the Composite

Notice that the alloys Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20) exhibit a caret-like entropy change curve, resulting in a relatively low cooling power, especially when the operating temperatures of the working temperature deviate from the transition temperature. Thus, it is important to prepare the composite with Gd and Gd1-xHox undergoing successive transition temperatures [34,35,36], where the composite owns two Curie temperatures and the entropy change curves exhibit two peaks which are located at the Curie temperatures of the two component materials, respectively. The composite material generally exhibits a gentler and large magnetic entropy change over a wide temperature region such that the δTFWHM as well as the RC and RCP will be enhanced. For this reason, we design and establish one new composite material composed with Gd and Gd0.80Ho0.20, The isothermal magnetic entropy change with different Gd mass ratios y in the composite are to be analyzed and discussed. Subsequently, the isothermal magnetic entropy change curves of the composite with different y (0.20, 0.40, 0.60, 0.67, 0.80) are obtained under 2 T field variation, as shown in Fig. 2. It is found that the optimized mass ratio yopt of Gd is equal to 0.67 by using the optimization method which is based on the following equations [23]:

$$ \hbox{-} \Delta {S}_{com}\left(\Delta B,{T}_{c, Gd}\right)=\hbox{-} \Delta {S}_{com}\left(\Delta B,{T}_{c,{Gd}_{0.80}{Ho}_{0.20}}\right) $$
(11)
$$ \hbox{-} \Delta {S}_{com}\left(\Delta B,{T}_{c, Gd}\right)=y\left[\hbox{-} \Delta {S}_{Gd}\left(\Delta B,{T}_{c, Gd}\right)\right]+\left(1-y\right)\left[\hbox{-} \Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\left(\Delta B,{T}_{c, Gd}\right)\right] $$
(12)
$$ \hbox{-} \Delta {S}_{com}\left(\Delta B,{T}_{c,{Gd}_{0.80}{Ho}_{0.20}}\right)=y\left[-\Delta {S}_{\mathrm{Gd}}\left(\Delta B,{T}_{c,{Gd}_{0.80}{Ho}_{0.20}}\right)\right]+\left(1-y\right)\left[-\Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\left(\Delta B,{T}_{c,{Gd}_{0.80}{Ho}_{0.20}}\right)\right] $$
(13)

where the entropy change of Gd and Gd0.80Ho0.20 is listed in Table 3.

Fig. 2
figure 2

The magnetic entropy change versus temperature curves of the composite with Gd and Gd0.80Ho0.20, where y denotes the mass ratio of Gd in the composite as well as 2 T applied magnetic field variation

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Table 3 The isothermal entropy change values of Gd and Gd0.80Ho0.20 at their respective Curie temperature (264.4 K and 293.0 K) under 2 T magnetic field variation
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It can be clearly seen from Fig. 2 that the isothermal entropy change curve of the composite with the optimized mass ratio of Gd is most gentle than that of the composites with other Gd mass ratios, where the maximal entropy change and δTFWHM reach 4.17 J/(kgK) and 87.3 K. Furthermore, the influence of Gd mass ratio y on the MCE and refrigeration characteristics of the composite including ΔSmax, δTFWHM, RCP, RC, Tr1, and Tr2 can be evaluated and discussed on the basis of Fig. 3. From Fig. 3a, one can see that Tr2 increases to a constant 305.2 K when y > 0.67. There is a region with rapid increased slope of Tr2 around y = 0.33. On the other hand, Fig. 3a shows that Tr1 firstly decreases and then increases with increasing Gd material mass ratio; its minimal value is 217.9 K when y = 0.67. Accordingly, when y = 0.67, δTFWHM of the composite material achieves its maximal value of 87.3 K, which is coincident with the optimized Gd material mass ratio calculated theoretically above. Additionally, it can be seen in Fig. 3b that the maximal entropy change exhibits a V-shaped curve with a minimal value 4.17 J/(kgK), while RC achieves a maximal value 280.6 J/kg when y = 0.67 and RCP curve exhibits a plateau, 360 J/kg or so, when y is between 0.33 and 0.67.

Fig. 3
figure 3

The Tr, Tr2, δTFWHM, maximal isothermal entropy change, and RC and RCP curves versus Gd mass ratio y in the composite under applied field variation from 0 to 2 T

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In Fig. 1, it can be seen clearly that the Curie temperature of Gd1-xHox decreases while the maximal entropy change increases with increasing Ho concentration in Gd1-xHox alloy. This means that the doping of Ho into Gd not only can adjust the Curie temperature but also can enhance the MCE of material. This is also matched with the following equation [37, 38]:

$$ \Delta {S}_{\mathrm{max}}=-1.07 qR{\left(\frac{g{\mu}_B JH}{kT_c}\right)}^{2/3} $$
(14)

where q is the number of magnetic ions and there is an inverse correlation between Curie temperature and the maximal entropy change of alloys. The decrease of material Curie temperature will lead to shifting Tr1 and Tr2 to lower temperature with a constant δTFWHM, and both RCP and RC increase with enhanced material entropy change, which can be found in Table 2.

In Fig. 2, the entropy change curves of the composite are gentler than that of single component material shown in Fig. 1. It can be ascribed by the fact that the entropy change of the composite equals the linear combination of the component material entropy change. Additionally, the component Gd-Ho alloy has only one entropy change peak at its Curie temperature, but the composite material owns more than one entropy change peak, whose number depends on the component material number. Moreover, the maximal entropy change of the composite will decrease to a certain extent, but it can enhance largely the δTFWHM, as seen in Table 4. This may be also explained as follows: The entropy change of the composite results from the linear combination of the component material entropy changes, that is,

$$ -\Delta {S}_{com}\left(\Delta B,T\right)=\left[-\Delta {S}_{Gd}\left(\Delta B,T\right)\right]y+\left[-\Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\left(\Delta B,T\right)\right]\left(1-\mathrm{y}\right) $$
(15)
Table 4 The MCE and refrigeration characteristics of the composite composed with Gd and Gd0.80Ho0.20 under 2 T applied magnetic field variation
Full size table

By integrating Eq. (15), it yields

$$ {\displaystyle \begin{array}{l}{\int}_0^{\infty }-\Delta {S}_{com}\left(\Delta B,T\right) dT\\ {}={\int}_0^{\infty }-\Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\left(\Delta B,T\right) dT+y{\int}_0^{\infty}\left\{\left[-\Delta {S}_{Gd}\left(\Delta B,T\right)\right]-\left[-\Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\right(\Delta B,T\left)\right]\right\} dT\end{array}} $$
(16)

In the case, one can see from Fig. 1 that the value of –ΔSGd(ΔB,T) is larger/smaller than those of \( \hbox{--} {\varDelta S}_{Gd_{0.80}{Ho}_{0.20}}\left(\varDelta B,T\right) \)when the temperature is larger/smaller than 275 K. And both –ΔSGd(ΔB,T) and \( \hbox{--} {\varDelta S}_{Gd_{0.80}{Ho}_{0.20}}\left(\varDelta B,T\right) \) approximately equal zero at ultra-high and low temperatures. Consequently, \( {\int}_0^{\infty}\left\{\left[-\Delta {S}_{Gd}\left(\Delta B,T\right)\right]-\left[-\Delta {S}_{Gd_{0.80}{Ho}_{0.20}}\right(\Delta B,T\left)\right]\right\} dT \) can be thought as a tiny value, and the \( {\int}_0^{\infty }-\Delta {S}_{com}\left(\Delta B,T\right) dT \) can be approximately thought as a constant. Thus, when the maximal entropy change decreases, the δTFWHM generally increases to some extent.

In Fig. 3a, one can see that the slope of Tr2 curve increases when y = 0.33. It can be explained in Fig. 4, where the entropy change of the composite at 264.4 K is equal to 5.24 J/(kgK) which is approximately twice of that at 293.2 K. This indicates that the temperature Tr2 changes dramatically when y is equal to 0.33. Furthermore, the δTFWHM of entropy change curve of Gd-Ho alloy reaches the largest value when Gd mass ratio equals to 0.67, which is consistent with the value 0.67 shown in Fig. 2. It means that the calculation method of the optimal mass ratio can optimize not only the component material mass ratio to get a gentler entropy change curve of the composite but the δTFWHM related to RC and RCP indirectly.

Fig. 4
figure 4

The magnetic entropy change curve of the composite composed with Gd and Gd0.80Ho0.20, where Gd mass ratio is 0.33

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There is a difference in Fig. 3b that the RCP exhibits a plateau while RC exhibits an obvious maximum. It can be explained with the definition of the related parameters. The RCP is determined by the maximal entropy change and the δTFWHM in the entropy change curve, while RC depends on the δTFWHM and the entropy change within the temperature region of the δTFWHM. Generally the value of RCP is larger than that of RC, and the values are the same if the entropy change curve would be absolutely flat from Tr1 to Tr2. Therefore, the ratio of RC to RCP can stand for the gentle degree of the related entropy change curve. The larger the ratio, the gentler the related entropy change curve is. For example, one can see from Fig. 3b that when y equals to 0.67, or the value of RC/RCP is equal to 0.772 shown in Table 4, the composite material entropy curve is gentlest.

5 Conclusions

Based on the de Gennes factor model method, the MCE of Gd, Gd1-xHox (x = 0, 0.05, 0.10, 0.15, 0.20), and the composite composed with Gd and Gd0.80Ho0.20 are investigated numerically. It is found that the composite composed of Gd and Gd0.80Ho0.20 with the mass ratio of 0.67:0.33 can reach the most perfect entropy change curve, with a maximal entropy change of 4.17 J/(kgK) and δTFWHM of 87.3 K, where Tr1 and Tr2 are 217.9 K and 305.2 K respectively under applied magnetic field varying from 0 to 2 T. Simultaneously, RC of the composite reaches the maximum value, 280.6 J/kg, when Gd mass ratio is 0.67, and RCP reaches a plateau, about 360 J/kg, when Gd mass ratio is between 0.33 and 0.67, under 2 T field change.