Magnetic skyrmions are axisymmetric topological solitons of vortex-like character hosted in ferromagnetic materials. Generally, they are stabilized by an exchange interaction of relativistic nature called Dzyaloshinskii–Moriya interaction (DMI) [1, 2]. Magnetic skyrmions are characterized by 1) a skyrmion number S (otherwise called the topological charge), an integer that indicates how many times magnetic moments within a skyrmion wrap a sphere; 2) helicity number, the phase appearing in the in-plane spin texture; and 3) a fixed rotation fashion called chirality χ. The skyrmion number is expressed as S = 1/(4π) ∫ d² ρ m·(∂m/∂x × ∂m/∂y) where m( ρ ) = M( ρ )/M _s is the dimensionless magnetization vector with M representing the magnetization, ρ = (x, y) the in-plane coordinates, M _s representing the saturation magnetization, and ∂/∂x and ∂/∂y are first partial derivatives.

Recently, great efforts have been devoted to the manipulation of magnetic skyrmions forming in magnetic nanostructures based on their ground-state magnetic properties and giving rise to spintronic applications [1–4]. Very recently, magnetic skyrmions have been employed as qubits, a new class of quantum logic elements [5]. It has also been proposed the employment of magnetic skyrmions as information entropy carriers suggesting a data communication system based on the coding of information entropy [6]. This investigation stemmed from the theoretical and numerical exploration of the thermodynamic properties of magnetic skyrmions [7–10]. In this respect, the link between the physical and information entropies has been a subject matter of several studies [11–15]. On the other hand, the concept of negentropy was introduced first by Szilard who solved Maxwell’s demon paradox [16, 17], and then by Brillouin [18–23] who continued Szilard’s and Shannon’s investigations. After the discovery and formulation of Landauer’s principle [24–27], in recent decades, great efforts were made for the full understanding of information erasure and its relation with thermodynamics and logical computation from a philosophical, theoretical, and experimental viewpoints [28–47].

In this perspective article, this kind of investigation was applied to magnetic skyrmions. The aim of this study was threefold: 1) to show that Landauer’s limit is expressed in terms of variation of negentropy for a Néel skyrmion; 2) to show that the generalized second principle of thermodynamics based on Brillouin’s negentropy principle can be applied to a Néel skyrmion; and 3) to explain the interplay between information and negentropy and vice versa of a Néel skyrmion [6, 8]. This discussion could open the route in prospect for a new way of storing and coding information by using magnetic topological defects.

A Néel skyrmion (or hedgehog skyrmion) forming in magnetic nanostructures as a result of the interfacial DMI is characterized by the magnetization texture m = χ ⁡ sin ⁡ θ   ρ ^ + cos ⁡ θ   z ^ in a cylindrical reference frame (ρ, φ, z) where χ = ±1 is the chirality (+1 outwardly magnetization, −1 inwardly magnetization), and θ is the polar angle with 0 ≤ θ ≤ π. Figure 1A shows an outwardly Néel skyrmion (χ = +1) with a negative polarity (θ = π) and skyrmion number S = −1 subjected to an external magnetic field H along the + z direction, while Figure 1B displays an inwardly Néel skyrmion (χ = −1) with a positive polarity (θ = 0) and skyrmion number S = +1 subjected to an external magnetic field H along the − z direction. In the following discussion, the Néel skyrmion texture shown in Figure 1A was taken into account to be consistent with the results of micromagnetic simulations carried out on a Néel skyrmion with χ = +1, negative polarity, and S = −1 [7]. However, note that this choice is purely arbitrary and the same conclusion would be drawn taking into account the Néel skyrmion with χ = −1, a positive polarity, and S = +1.

The skyrmion energy was calculated from the microscopic micromagnetic Hamiltonian as a spatial integral of the skyrmion energy density within the thin-film limit including exchange, interfacial DMI, magnetostatic and perpendicular anisotropy contributions, and external magnetic field interaction [7–9]. Within this model, the exchange interaction among spins forming the magnetic skyrmion was rigorously taken into account. It was found that, in the vicinity of the absolute energy minimum at the equilibrium skyrmion diameter D _0sky, the skyrmion energy can be fitted by means of a parabolic curve for any temperature T and bias field amplitude H in the region of skyrmion metastability (0 ≤ T ≤ 300 K for μ ₀ H > 5 mT and 0 ≤ T ≤ 200 K for μ ₀ H = 0 mT) [7]. Importantly, D _0sky strictly depends on the parameters of the microscopic Hamiltonian. In this respect, the determination of the skyrmion size for an isolated skyrmion by computing the skyrmion radius (both equilibrium and average) has recently been proved according to different analytical theories based on the minimization of the skyrmion energy with respect to the skyrmion radius [9, 48, 49]. In particular, it has been shown that both the average skyrmion size and the wall width separating the core from the outer domain of the skyrmion can be accurately computed [48]. This investigation has been generalized by studying the magnetic skyrmion’s size and spin profile in a condensed phase forming a skyrmion crystal at high skyrmion density [49]. In this latter case, it has been demonstrated that the dependence of skyrmion size on magnetic parameters is different compared to isolated skyrmions or to skyrmion stripes forming at low skyrmion density.

According to micromagnetic simulations, it was observed that the value of the Néel skyrmion diameters obeys a distribution analogous to Maxwell–Boltzmann (MB) of the molecules of an ideal gas at any T and for any H in the region of metastability [7]. Exploiting this physical analogy with ideal gases, an analytical MB distribution for a 3D skyrmion diameter population [7, 9] was proposed, and it was found an excellent agreement between the micromagnetic and the analytical results [7]. This analogy was also extended to the 2D skyrmion diameter distribution [8]. From the analogy with the 3D MB distribution of an ideal gas, the Gaussian distribution at the thermodynamic equilibrium at a given T and H for both 3D and 2D skyrmion diameter distribution can be written in the form f 0 ( D sky ) = C av   e −   a k B T ( Δ 〈 D sky 〉 ) 2 , (1) where C _av is the normalization constant (in m^-2), k _B is the Boltzmann constant, a is a coefficient proportional to the skyrmion energy curvature, Δ<D _sky> = D _sky − < D _sky >, D _sky the skyrmion diameter, and <D _sky> the average skyrmion diameter with <D _sky> = <D _sky (T)>.

Owing to the mentioned analogy, it is useful to relate the diameter distribution depending on T and H to skyrmion’s thermodynamic entropy as occurs for the thermodynamic entropy of an ideal gas. Regarding this, it is important to note that the main source of entropy for domains forming in ferromagnets is represented by spin waves (or magnons). Recently, it has been found that the source of entropy and free energy for a domain and a domain wall (DW) in a magnetic nanowire is due to thermally activated magnons [50]. In this system, it has been demonstrated that the larger domain wall entropy is due to the increase in the magnon density of states at low energy, and the driving force allowing DW propagation under a temperature gradient towards the hotter region is the thermodynamic entropy itself. Under this condition, the system evolves toward a state that lowers its free energy by exploiting DW’s larger entropy [50]. The DW movement toward a hotter region driven by thermal gradients has also been proved in antiferromagnets and can be understood by means of the minimization of the free energy [51]. Also, the main source of the configurational entropy of a classical Néel magnetic skyrmion has been attributed to the thermal-breathing mode, a type of spin wave as observed in micromagnetic simulations [7].

The configurational entropy at thermodynamic equilibrium related to a classical Néel magnetic skyrmion diameter distribution was computed, at each T and H, as the Gibbs–Boltzmann’s statistical thermodynamic entropy, a quantity proportional to the statistical average H ₀ = <lnf ₀>, the Boltzmann order function at thermodynamic equilibrium, namely as S = −k _B H ₀ with S = S(T) [7–9]. This entropy is the generalization of the Boltzmann entropy when the microstates of the statistical ensemble are not equiprobable. For a 2D skyrmion diameter population, after performing the statistical average H ₀ within the continuous limit, it takes the form [8] S = − k B π ∫ 0 ∞ d D sky D sky f 0 ( D sky ) ln ( f 01 ( D sky ) ) . (2)

Here, f ₀₁ (D _sky) = f ₀<A _sky> with <A _sky> ≈1/4 π <D _sky>² the average skyrmion area and S > 0 (in J/K). The Gaussian distribution f ₀ is the one that realizes the largest thermodynamic entropy according to the maximum entropy principle. In turn, <D _sky (T)> ≈ D _0sky [1+ k _B T/(2a D _0sky ²)] with D _0sky = D _0sky(T) defined as the diameter at which the total skyrmion energy attains its absolute minimum. In turn, the value of D _0sky strictly depends on the magnetic parameters appearing as coefficients in the micromagnetic Hamiltonian such as the exchange stiffness constant A, the interfacial Dzyaloshinskii–Moriya parameter D, the uniaxial perpendicular anisotropy constant K _u, and on the external magnetic field amplitude H. The statistical thermodynamic entropy is also referred to as thermal entropy [52, 53] and is an increasing monotonic function of T.

The information entropy (expressed in terms of the number of bits) was calculated according to Jaynes’s information framework [11–15] and taking into account Eq. 2. The use of continuous variables was suggested by Jayne in [13, 14] and was applied to the definition of information entropy in the continuum case [15]. The information entropy (in bits) was determined as the 2D statistical average of the information content I (D _sky) = −log₂ (f ₀₁ (D _sky)) (2 is the logarithm basis) [6] and can be rewritten in the form H I = π ∫ 0 ∞ d D sky D sky f 0 ( D sky ) I ( D sky ) , (3) with H _I > 0.

Landauer’s limit is derived starting from the configurational entropy and the corresponding information entropy. To create bits of information, S must decrease passing from an initial temperature T _i to a final temperature T _f with T _f < T _i. Starting from Eq. 1 and using some logarithm rules, the entropy variation ΔS = S (T = T _f) − S (T = T _i) with S (T = T _f) < S (T = T _i) such that ΔS < 0 can be written in a compact form as Δ S = k B π [ ∫ 0 ∞ d D sky D sky ⁡ ln ( f 01 ( T = T i ) f 0 ( T = T i ) f 01 ( T = T f ) f 0 ( T = T f ) ) ] , (4) where the dependence of f ₀₁ and f ₀ on D _sky is omitted. Analogously, the variation of the information entropy coded by the magnetic skyrmion, ΔH _I = H _I (T = T _f) − H _I (T = T _i) with T _f > T _i (the temperature T _f appearing in H _I corresponds to the temperature T _i appearing in S and vice versa) and H _I (T = T _f) > H _I (T = T _i) such that ΔH _I >0 takes the form Δ H I = π [ ∫ 0 ∞ d D sky D sky ⁡ log 2 ( f 01 ( T = T i ) f 0 ( T = T i ) f 01 ( T = T f ) f 0 ( T = T f ) ) ] . (5)

It is useful to introduce the corresponding thermodynamic variation of information entropy by defining S _I = k _B H _I . By comparing Eqs 4, 5 and taking into account that S _I = k _B H _I , Landauer’s limit can be derived. Indeed, substituting ln (f ₀₁) = log₂ (f ₀₁) ln2 in Eq. 2 and comparing Eq. 2 with Eq. 3, one gets, via ΔS _I = k _B ΔH _I , ΔS = −ΔS _I ln2. The creation of 1 bit of information leads to a variation ΔH _I = 1 bit and to an increment ΔS _I = k _B (units of 1 bit) yielding Δ S = − k B ⁡ ln ⁡ 2. (6)

Therefore, Landauer’s limit corresponds to the lower limit of the entropy lost in an observation as a result of the creation of 1 bit. In the present case, the thermodynamic entropy was lowered to create 1 bit of information and the minimum energy, E = − k B T   ln ⁡ 2 , (7) was subtracted from the system. The amount of energy is the minimal work W = − k _B T ln 2 (W < 0) that must be extracted to create 1 bit of information, as established by Landauer’s principle. This is a different case with respect to that was considered by Landauer for which the logical irreversibility implies the thermodynamic irreversibility. The information coding by a magnetic skyrmion can be regarded as a thermodynamically reversible process. By introducing the negentropy, the entropic equivalent of degradation of energy [18], namely N = −S (N <0 and S >0) and ΔN = −ΔS >0 being ΔN = N (T = T _f) − N (T = T _i), N (T = T _f) < 0 and N (T = T _i) < 0 but N (T = T _f) > N (T = T _i) we get Δ N = k B ⁡ ln ⁡ 2. (8)

Therefore, Landauer’s limit can be regarded as the negentropy acquired by the system.

According to the second principle of thermodynamics, a magnetic system moves towards a state with a larger entropy or lower free energy [50]. The magnetic skyrmion’s Helmholtz free energy, viz. F = <E > − T S with <E > = <E (T)> the average skyrmion energy, in the absence of an external magnetic field, diminishes with increasing T and attains a minimum at the upper limit of the region of metastability at T = 300 K. With the magnetic parameters used (see the following section), F ≈ 5.5 10^–20 J at T = 150 K [9] corresponds to 1 bit of information.

Here, the generalized second principle of thermodynamics for a closed thermodynamic system such as a magnetic skyrmion in terms of Brillouin’s negentropy principle ΔS _tot = ΔS − ΔS _I ≥ 0 is discussed [18]. Therefore, the total entropy S _tot of a magnetic skyrmion does not decrease. In particular, ΔS _tot = S _{tot f} − S _{tot i} is the total entropy variation from the initial state i to the final state f, and ΔS = S _f −S _i >0 is the variation of the thermal entropy from the initial state to the final state (by the convention of an opposite sign with respect to that in Eq. 4), while ΔS _I = S _{I f} −S _{I i} > 0 is the increment of information entropy in thermodynamic units owing to the creation of bits of information. By introducing the negentropy variation ΔN = −ΔS < 0, the generalized second principle of thermodynamics is expressed in terms of Brillouin’s negentropy Δ ( N + S I ) ≤ 0 . (9)

Eq. 9 expresses thermodynamic reversibility when Δ(N + S _I ) = 0, viz. ΔN = −ΔS _I but it does not state that physical reversibility necessarily implies logical reversibility.

In this section, new perspectives in information theory based on the use of the magnetic skyrmion as a unit of information entropy are outlined [6, 8].

In this study, the interplay between thermodynamics and information occurring in easily manipulated magnetic skyrmions forming in magnetic nanostructures was discussed. It has been proved that Landauer’s limit for a magnetic skyrmion can be expressed in terms of negentropy variation. It has been shown that the interchange between thermodynamic entropy and information entropy to create bits of information occurs by using a reservoir of negentropy that fulfills Brillouin’s negentropy second principle of thermodynamics. This type of coding information based on the information entropy could be employed in prospect for improving data transmission.