1. Introduction
The magnetocaloric effect (MCE) represents the basis for attractive refrigeration technologies for commercial and cryogenic applications [
1,
2,
3,
4] and also for better understanding of the magneto-physical properties of magnetic materials. Most studies in this area are devoted to the creation of efficient refrigerants operating at nearly room temperature; however, refrigerants functioning at low and ultra-low temperatures [
1,
3,
4] are an important as well as cost-effective alternative to dilution refrigerants using a mixture of
3He and
4He isotopes. Recently, the problem of finding low-temperature coolants has become increasingly important in connection with the development of quantum computers requiring cryogenic conditions for operating. While magnetically ordered crystals are promising as coolants operating at room temperatures, molecular nanomagnets can act as coolants, demonstrating MCE at low temperatures [
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23]. To date, extensive research devoted to MCE has been carried out. Polynuclear high-spin clusters of 3d metals [
11,
12,
13,
14,
15,
16,
17,
18,
19], lanthanide-based clusters [
20,
21], and mixed 3d-4f structures of various topology have been shown to be promising as magnetic coolants [
22,
23]. Some dimeric and monomeric 4f-based compounds exhibit significant MCE [
24,
25,
26,
27]. The mononuclear 3d metals-based systems with spin
S = 5/2 [
28],
S = 3/2 [
29], and
S = 1 [
30,
31], which possess little anisotropy, have been shown to exhibit relatively high MCE. Important advances in the field of MCE have been achieved by using crystalline systems. Very recently, the low-temperature magnetocaloric responses in solid-state magnets have been extensively studied with the aim to develop high-performing MC materials for magnetic refrigeration and in-depth understanding of their magneto physical characteristics [
32,
33,
34]. Thus, in ref. [
32], a remarkable low-temperature magnetocaloric response in GdNaGeO
4 oxide was discovered, and this system was found to be an excellent candidate for application in low-temperature magnetic refrigeration. The influence of substitutions (Fe, Mn, Cu, and Al) on the magnetic properties and magnetocaloric effect of the GdCo
2 compounds was revealed in ref. [
33]. Finally, the new material Gd
11O
10(SiO
4)(PO
4)
3 for cryogenic magnetic refrigeration application was synthesized, and its structure, magnetic properties, and cryogenic magnetocaloric performances were studied in detail [
34].
Noting the achievements reached in the field of solid-state magnetic resonance, it is also necessary to mention the undoubted advantages of nanomagnets, which include the possibility of the chemical control of their structure and properties (especially magnetic anisotropy [
5,
6,
7]) to achieve maximum effect. Although high anisotropy is generally considered undesirable, highly anisotropic compounds are ideal candidates for observing the so-called rotational MCE [
8,
9], where the field can be effectively modulated by rotating the sample. This method not only allows the use of highly anisotropic clusters but also enables rapid switching of the field when needed [
10].
Recently, we analyzed the magnetothermal processes expected in mononuclear magnetically anisotropic paramagnetic complexes of 3d ions [
35], which are induced, as in the conventional MCE, by changing the magnetic field. However, unlike traditional approaches in the MCE area, we focused on the case when the field is switched off “suddenly”, which leads to a violation of the thermodynamic equilibrium between different parts of the system. A similar approach has been reported in refs. [
36,
37] dealing with the modeling of non-equilibrium magnetothermal processes in Heisenberg-type and Ising-type dimers. The classical definition of MCE assumes that quasi-equilibrium reversible processes generated by the relatively slow change of the magnetic field do not involve the spin–lattice relaxation. Although, according to the adopted terminology, the magnetothermal effect considered in [
35,
36,
37] cannot be referred to as MCE, these two kinds of effects are somewhat close in nature. The approach developed in refs. [
35,
36,
37] assumes the experimental conditions of the efficient heat exchange between the magnetic molecule and its surroundings so that the system maintains the same temperature at the end of the relaxation process as that which existed before the field switching off. The quantitative measure of such process is the heat release, which is the thermal energy passing as heat from the spin subsystem composed of anisotropic mononuclear 3d metal complexes to the phonon bath and further on to the surrounding area or from the surrounding area to the phonon bath and then to the spin subsystem depending on the sign of the thermal effect.
An even more interesting experimental situation may occur when the sample is isolated from its surrounding with an adiabatic envelope. Under this condition, sudden magnetic field switching off is expected to lead to increasing or decreasing temperature depending on the sign of the thermal effect. To evaluate the temperature change, the theoretical approach developed in [
35,
36,
37] should be essentially modified. A new approach and its application to the analysis of the thermal effects are discussed in this article.
2. Theoretical Approach
We consider a mononuclear 3d metal complex with spin
S = 1 (e.g., high-spin Ni(II) complex) exhibiting axial magnetic anisotropy. Magnetic anisotropy in magnetic materials and metal clusters has long been studied in connection with the EPR, and this fundamental property of magnetic materials has become even more relevant in the development of field of single molecular magnets (for a detailed overview, see ref. [
38]).
Such a complex is described by the following spin-Hamiltonian:
The first term in Equation (1) is the axial zero-field splitting (ZFS) operator, where
D is the axial ZFS parameter (for a detailed overview, see ref. [
38]),
is the spin-projection operator, and
Z is the anisotropy axis. The second term represents the Zeeman interaction, where
is the external magnetic field, which is applied along the
Z axis;
is the axial
g-tensor component;
is the Bohr magneton. Depending on the sign of the parameter
D, the system can exhibit either easy-axis-type magnetic anisotropy (
D < 0) or easy-plane-type anisotropy (
D > 0).
Although we consider only a fairly simple S = 1 system to illustrate the background of the proposed concept, the main results seem to extend to complexes possessing arbitrary spin values. In fact, the basic Hamiltonian describing anisotropy, Equation (1), is applicable to any spin value, while the remaining evaluations for the high spin values remain the same, but of course, the expressions become more complex. In particular, the expressions for the energy levels, Equation (2), yield simple generalization. At the same time, for , the spin-Hamiltonian involves high-order terms (), but the corresponding contributions are usually small and should be taken into account when the experimental data (for instance, EPR) clearly indicate their presence.
The eigenvalues of the Hamiltonian, Equation (1), are the following:
Figure 1 shows the field dependences of the eigenvalues, Equation (2), evaluated for two signs of
D. It is seen that at
D < 0 (
Figure 1a), the zero-field ground state of the system is the doublet
, while the excited state is the singlet
. In the field directed along the anisotropy axis, the ground doublet is split in such a way that the Zeeman sublevel with
proves to be the ground state. This is a case of an easy-axis-type magnetic anisotropy. At
D > 0, the
—state is the ground one at zero field (
Figure 1b). This state is non-magnetic, so we deal with the easy-plane-type anisotropy. In this case, at non-zero field, the ground state depends on the ratio between the Zeeman energy
and the ZFS parameter
D. In the case of a weak field, i.e.,
, the ground state possesses
, while for a strong field, i.e.,
, the Zeeman sublevel with
becomes the ground state. Below, we show that this difference in the energy spectra related to the cases of
D < 0 and
D > 0 determines the difference in the thermal effect (cooling versus heating) caused by fast magnetic field switching off. It is also demonstrated that the difference between the weak and the strong field cases in systems with
D > 0 proves to be important, although the thermal effect does not depend on the field.
It is convenient to set the zero-field level
as a reference energy so that the eigenvalues of the Hamiltonian, Equation (1), acquire the following form:
Let us assume that at the beginning of the process, the system is exposed to the magnetic field
B directed along the anisotropy axis
Z, and it is in a thermodynamic equilibrium state characterized by temperature
T. Taking into account Equation (3), one obtains the following expressions for the Boltzmann populations of the spin states:
where
kB is the Boltzmann constant, and the denominator in these expressions represents a partition function. At fast field switching off, the energy exchange between the spin subsystem and the thermal bath does not have time to occur, so the same populations also remain immediately after field switching off. However, these populations reach non-equilibrium because the energy pattern is changed when the field is switched off. It is also notable that the temperature of the phonon subsystem does not change in the course of the switching off event and remains equal to the temperature
T, which characterizes the initial (before the field switching off) equilibrium state.
In course of the subsequent relaxation process, the system comes to the new equilibrium state characterized by the final temperature
Tfin. Immediately after the field is switched off, the internal energy of the spin subsystem is given by the sum of the zero-field spin-state energies weighted by their populations defined in Equation (4):
where
is the Avogadro constant, and the symbol
init is used to specify the internal energy of the spin subsystem at the beginning of the relaxation process. With the adopted choice of the reference energy, i.e.,
one obtains
In the subsequent relaxation process, the heat exchange occurs between the spin subsystem and the phonon bath, in course of which the final (
fin) equilibrium state is established. This final state is characterized by the following Boltzmann populations of the zero-field states:
This allows us to write down the following expression for the final internal energy of the spin subsystem:
The thermal balance in the entire system, including spin subsystem and phonon bath, is maintained provided that
where
is the heat capacity of the phonon subsystem, which comes from the acoustic phonons. It is assumed that the temperature is low enough that the optical phonon branches are not excited. In this particular but important case, the phonon heat capacity can be described by the Debye law:
where
is the Debye temperature. The number of oscillating units in Equation (10) is assumed to be the same as the number of spins in Equation (5), which seems to be a good approximation as applied to the long-wave acoustic phonons participating in phonon heat capacity, when each spin complex can be imagined to be oscillating as a whole. Then, one obtains
Substitution of Equations (6), (8) and (11) into Equation (9) results in the following final equation for
:
By solving this equation, we find as a function of the initial temperature T and also the temperature change as a function of T.
The approach described so far can be reformulated in terms of the so-called “spin temperature”. The latter can be used to describe a nonequilibrium state in certain systems, particularly those with strong spin–lattice interactions. While equilibrium states are characterized by a uniform temperature throughout the volume, nonequilibrium states can exhibit spatially varying temperatures or a temperature different from the temperature of a system in equilibrium with its environment.
Within this approach, it is assumed that the nonequilibrium state of the spin subsystem, formed immediately after the field is switched off, can be treated as a quasi-equilibrium state, with the term “equilibrium” referring exclusively to the spin subsystem and not to the entire system. This state is characterized by a spin temperature
that differs from the temperature of the phonon subsystem (the latter remains unaffected by the rapid field switching off and retains its initial temperature
T). The initial internal energy of the spin subsystem is then expressed through the spin temperature as follows:
By equating Equations (6) and (13), one obtains the following expression for the spin temperature:
Now, the thermal balance condition can be written as follows:
where the value
is the heat capacity of the spin subsystem. We thus obtain
By substituting Equations (16) and (11) into Equation (15), one arrives at the following final equation:
It should be emphasized that Equation (17) is equivalent to Equation (12) provided that is defined by Equation (14).
3. Results and Discussion
Figure 2 and
Figure 3 show the series of the dependencies of the temperature change
of the system induced by fast magnetic field switching off on the initial temperature
T. The results are shown for various negative (
Figure 2) and positive (
Figure 3) values of
D and various values of
θD and
B, provided that the system is isolated from the environment by an adiabatic shell. In all calculations, it is assumed that
, which is in the typical range for Ni(II) complexes.
It follows from
Figure 2 that at
D < 0, the temperature change
is negative (the temperature is decreased) at finite values of
T, which means that sudden magnetic field switching off leads to the magnetic cooling in system with easy-axis-type magnetic anisotropy. This magnetic cooling disappears in the limits of low and high temperatures and reaches maximum at a finite temperature corresponding to the minimum in
vs.
T curve. By comparing curves in
Figure 2a,b, one can see that decrease in the temperature is higher for larger ratio
B/
D, and its maximum is shifted towards higher
T with the increase in this ratio (compare the curves in
Figure 2a,b). It is also seen that the cooling effect is stronger for systems with higher
θD (compare
Figure 2a,c).
To better understand this behavior, let us focus on
Figure 4, in which the temperature dependencies of the nonequilibrium populations of the energy levels (established immediately after the magnetic field is turned off) are compared with the corresponding equilibrium populations.
Figure 4a shows these populations evaluated in the case of
D < 0. The nonequilibrium populations of the ground and excited levels in this case are expressed as follows:
where
and
are the nonequilibrium populations of the states defined by Equation (4). Similarly, the equilibrium populations of the levels are the following:
where the equilibrium states populations are given by
It follows from
Figure 4a that in the case of
D < 0, the excited level with
MS = 0 proves to be underpopulated, while the ground level with
MS = ±1 is overpopulated, which means that in the course of subsequent spin–lattice relaxation, the heat is absorbed from the phonon subsystem, initially having the temperature
T, to the spin system. As a result, the final temperature
Tfin established in the system upon relaxation proves to be lower than
T, as shown in
Figure 2. It is also seen from
Figure 4a that the difference between the equilibrium and nonequilibrium populations is vanishing in the low- and the high-temperature limits and passes through the maximum at some finite temperature. Comparing
Figure 4a with
Figure 2, one can see that just the maximal difference between the populations gives the greatest decrease in temperature.
In contrast, in the case of positive
D values,
is always positive at finite temperatures; i.e., sudden field switching off causes heating of the system (
Figure 3), with the heating effect being stronger for larger
B/
D and/or higher
θD. As distinguished from the case of
D < 0, for
D > 0, the shape of the
curve proves to be dependent on whether
< 1 (weak field) or
> 1 (strong field). It is seen from the plots in
Figure 3 that at weak field,
is vanishing in the low-temperature limit, while at the finite temperatures, it increases with the increase in
T, passes through the maximum, and finally decreases with further increase in
T. On the other hand, at strong field, the low-temperature limit of
tends to the nonzero value. In the latter case, the low-temperature limit of
Tfin is shown to satisfy the following equation:
where
. The increase in
T in the case of strong field leads to a monotonic decrease in
, as follows from the high-field plots in
Figure 3 (see, e.g., the curves with
B = 8 T and 6 T in
Figure 3a).
The described features of the thermal process in the case of
D > 0 can be understood by examining
Figure 4b,c, which compare the equilibrium and nonequilibrium populations evaluated for this case. When
D is positive, the equilibrium and nonequilibrium populations of the ground and excited levels are defined by Equations (22) and (23), respectively:
The nonequilibrium and equilibrium populations of the states, presented in the right parts of Equations (22) and (23), are given by Equations (4) and (20), respectively.
In the case of a weak field (
Figure 4b), one obtains
and
, which means that fast magnetic field switching off creates a nonequilibrium state with an overpopulated excited level (
MS = ±1 level,
Figure 1b), while the ground level with
MS = 0 is underpopulated. As a result, the heat is transferred from the spin subsystem to the phonon bath in course of the relaxation process, which leads to an increase in the temperature of the system relative to the initial temperature
T. The difference in the equilibrium and nonequilibrium populations disappears in the low- and the high-temperature limits (
Figure 4b), and hence, heating in these limits disappears as well.
The conditions
and
are also fulfilled for a strong magnetic field (
Figure 4c), which is indicative of the heating process in this case. However, as distinguished from the above-considered cases in which the difference between the equilibrium and nonequilibrium populations of each level disappears in the low-temperature limit, now, this difference is maximal at
T = 0 K, and it monotonically decreases with the increase in
T. This explains the monotonic decrease in
in the case of
D > 0 and strong magnetic field, as described above (see, e.g., the curves with
B = 8 T and 6 T in
Figure 3a). Another notable feature of this case is the occurrence of inverted populations at low temperatures. In the high-temperature limit, the equilibrium population reaches its maximum value of 2/3 for the excited doublet (
MS = ±1) and minimum value of 1/3 for the non-degenerate ground state (
MS = 0). Population inversion occurs when the excited doublet level population exceeds 2/3 and falls for the singlet ground level below 1/3. For
D = 5 cm
−1 and
B = 8 T (
Figure 4c), these conditions are satisfied provided that
T < 7 K.
Since the spin subsystem is effectively hotter in the nonequilibrium state with inverted population than in any state without it, the system exhibits greater heating when inversion occurs (compare the curves at high fields, where inversion takes place, with those at low fields in
Figure 3).
The described features of the thermal processes can be also understood on the basis of the spin-temperature formalism introduced at the end of
Section 2.
Figure 5 shows the spin temperature
Tspin evaluated as function of
T for the cases of negative (
Figure 5a) and positive (
Figure 5b,c)
D and different values of the magnetic field. It is seen that at
D < 0, the spin temperature is lower than
T (
Figure 5a), which is the consequence of underpopulation of the excited level (
Figure 4a). As a result, the heat is transferred from the phonons that initially have the temperature
T to the spin subsystem whose spin temperature is lower than
T, and hence, the final temperature
Tfin established in course of the relaxation process proves to be lower than the initial temperature
T. This is in accord with the negative sign of Δ
T (see
Figure 2). The difference
T–
Tspin is the increasing function of
B (
Figure 5a), which explains that cooling is more efficient for a higher magnetic field (compare, e.g., the
curves in
Figure 2a found for different
B-values).
In contrast, at a positive
D and weak field, the spin temperature is higher than
T (
Figure 5b) due to overpopulation of the excited level (see
Figure 4b), so the heat is transferred from the spin subsystem to the phonons. As a result, the final temperature of the entire system is higher than the initial temperature
T; that is, the system heats up.
Figure 5b shows that the difference between
Tspin and
T increases with the increase in
B, giving rise to stronger heating for a stronger initially applied field (see the plots in
Figure 3, which are related to the weak field case). Finally, for
D > 0 and a strong magnetic field, there are two ranges of
T exhibiting different signs of
Tspin (
Figure 5c). Below some critical temperature
Tc, the spin temperature becomes negative (
Figure 5c), indicating population inversion at low
T. For
T >
Tc, the spin temperature is positive and exceeds the lattice temperature,
Tspin >
T, as in the weak field case shown in
Figure 5b.
The critical temperature
Tc at which the spin temperature changes the sign is the temperature at which
and
. It follows from
Figure 5c that
Tc increases with the increase in the field. As indicated above, in the case of equilibrium distribution, such populations occur in the high-temperature limit. Then, the inverted populations (at
and
,
Figure 4c) occurring at
T <
Tc correspond to the negative spin temperature. When
T approaches
Tc from the left,
Tspin → −∞, while if
T approaches
Tc from the right,
Tspin → +∞. Since the negative absolute temperatures are actually “higher” than all positive temperatures, including “infinite temperature”, in the case of
Tspin < 0, the heat is also transferred from the spin subsystem to the phonons, giving rise to heating of the system (
> 0) in agreement with those plots in
Figure 3, which are related to the strong field case.