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Thermally induced magnetic relaxation in square artificial spin ice

View Article: PubMed Central - PubMed

ABSTRACT

The properties of natural and artificial assemblies of interacting elements, ranging from Quarks to Galaxies, are at the heart of Physics. The collective response and dynamics of such assemblies are dictated by the intrinsic dynamical properties of the building blocks, the nature of their interactions and topological constraints. Here we report on the relaxation dynamics of the magnetization of artificial assemblies of mesoscopic spins. In our model nano-magnetic system - square artificial spin ice – we are able to control the geometrical arrangement and interaction strength between the magnetically interacting building blocks by means of nano-lithography. Using time resolved magnetometry we show that the relaxation process can be described using the Kohlrausch law and that the extracted temperature dependent relaxation times of the assemblies follow the Vogel-Fulcher law. The results provide insight into the relaxation dynamics of mesoscopic nano-magnetic model systems, with adjustable energy and time scales, and demonstrates that these can serve as an ideal playground for the studies of collective dynamics and relaxations.

No MeSH data available.


Related in: MedlinePlus

Stretched exponentials and fitting of MTRM(T) and τ(T).(a) The variation of a stretched exponential, M = M0 exp[−(t/τ)β], for different values of β using a relaxation time τ = 1000 s. The black dots correspond to the recorded relaxation data for the d = 420 nm array at 200 K. In the inset the same data is fitted to a stretched exponential with β ≈ 0.15 and τ ≈ 1000 s, in the time window 10−10 (0.1 ns) to 1010 s (≈300 years) which covers the majority of the relaxation from M0 to zero magnetization. The magnetization is normalized to M0, which is the magnetization directly after the field is switched off (t = t0), corresponding to a fully dressed state of the array, see Fig. 1 (I). (b) A fit of the temperature dependence of the relaxation time τ to the Vogel Fulcher law, τ = τ0 exp(EB/kB(T − T0)), for both arrays, yielding an average energy barrier of EB/kB = 4500 K.
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f5: Stretched exponentials and fitting of MTRM(T) and τ(T).(a) The variation of a stretched exponential, M = M0 exp[−(t/τ)β], for different values of β using a relaxation time τ = 1000 s. The black dots correspond to the recorded relaxation data for the d = 420 nm array at 200 K. In the inset the same data is fitted to a stretched exponential with β ≈ 0.15 and τ ≈ 1000 s, in the time window 10−10 (0.1 ns) to 1010 s (≈300 years) which covers the majority of the relaxation from M0 to zero magnetization. The magnetization is normalized to M0, which is the magnetization directly after the field is switched off (t = t0), corresponding to a fully dressed state of the array, see Fig. 1 (I). (b) A fit of the temperature dependence of the relaxation time τ to the Vogel Fulcher law, τ = τ0 exp(EB/kB(T − T0)), for both arrays, yielding an average energy barrier of EB/kB = 4500 K.

Mentions: The relaxation in Fig. 3 occurs across a large time window and it is not described by a simple exponential function. However, a stretched exponential, M = M0 exp[−(t/τ)β], where β is the stretching exponent, M0 is the magnetization at t = 0 and τ is the relaxation time, can be used to describe the relaxation process1318. To illustrate the influence of β, the stretched exponential decay in the time window 0.1 to 10 000 s is plotted in Fig. 5a, using τ = 1000 s and M0 = 1. As can be seen in Fig. 5a, a simple exponential (β = 1) decays rapidly and does not describe the relaxation behavior illustrated in Fig. 3. In Fig. 5a the relaxation data measured at 200 K is shown for the 420 nm array; comparing this to the stretched exponential curves one observes that it accurately matches the β = 0.15 curve. The 200 K data was also fitted to a stretched exponential decay, yielding τ ≈ 1000 s and β ≈ 0.15; red line in the inset of Fig. 5a. Extending this fitting procedure to the measured relaxation data, presented in Fig. 3a,b, the temperature evolution of τ can be studied. It should be noted that τ and β are strongly coupled and therefore physical meaningful fits can only be made in a relatively narrow temperature region (see supplementary material for more details). For a region of about 30 K, β is relative stable (around 0.15 for both arrays) and a reliable determination of the temperature dependence of τ can be made. For example, τ changes by 3 orders of magnitude from about 105 to 102 s for both arrays, in the interval 180 to 210 K for the d = 420 nm array and 200 to 230 K for the d = 380 nm array. The extracted τ(T) values were fitted to the Vogel-Fulcher law1920, τ = τ0exp[EB/kB(T − T0)], where τ0 = 10−11 s is a constant that describes the relaxation time of the individual islands at very high temperatures, T0 describes the interaction strength, EB is the energy barrier which the magnetic moment of the island has to overcome in order to reverse its direction and kB is the Boltzmann constant. The fits for both arrays are shown in Fig. 5b. From these fits, the energy barrier, EB, and the interaction strength T0, are derived. A value of EB/kB = 4650 K is received for the 380 nm array and 4430 K for the 420 nm array, while for T0 the values are 72 and 62 K, respectively. The results for T0 are consistent with a stronger interaction between the elements in the 380 nm array as compared to the elements in the 420 nm array, as expected. The energy barrier of the islands is expected to be rather similar for the two arrays, due to the fact that the islands have the same size and shape in the two arrays and the temperature ranges for the fits partly overlap. Therefore fits where the energy barrier was fixed to 4500 K were also made, yielding T0 = 77 K for the 380 nm array and T0 = 60 K for the 420 nm array. The individual blocking temperature, TB, of an island can be determined from τobs = τ0exp[EB/kBTB], where τobs is the observation time and is taken as 30 s. Using EB/kB  =  4500 K a blocking temperature of approximately 160 K can be estimated. The energy barrier, EB, of the individual islands can mainly be attributed to their shape anisotropy. Using this approximation the energy barrier can be estimated from EB(T) = μoMs(T)2ΔNV/2, where μo is the vacuum magnetic permeability, Ms(T) is the magnetization of an island at temperature T, ΔN is the differential demagnetizing factor calculated using the Osborn methodology21 (see Supplementary materials for details), and V is the volume of the magnetic island. By using this approach and the temperature dependence of M described in the Methods, a mean value of the energy barrier was determined to be EB/kB = 4900 K at 195 K and EB/kB = 4500 K at 215 K. By comparing these calculated values of EB with the values estimated from the relaxation experiments, it can be seen that they are in good agreement.


Thermally induced magnetic relaxation in square artificial spin ice
Stretched exponentials and fitting of MTRM(T) and τ(T).(a) The variation of a stretched exponential, M = M0 exp[−(t/τ)β], for different values of β using a relaxation time τ = 1000 s. The black dots correspond to the recorded relaxation data for the d = 420 nm array at 200 K. In the inset the same data is fitted to a stretched exponential with β ≈ 0.15 and τ ≈ 1000 s, in the time window 10−10 (0.1 ns) to 1010 s (≈300 years) which covers the majority of the relaxation from M0 to zero magnetization. The magnetization is normalized to M0, which is the magnetization directly after the field is switched off (t = t0), corresponding to a fully dressed state of the array, see Fig. 1 (I). (b) A fit of the temperature dependence of the relaxation time τ to the Vogel Fulcher law, τ = τ0 exp(EB/kB(T − T0)), for both arrays, yielding an average energy barrier of EB/kB = 4500 K.
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC5121627&req=5

f5: Stretched exponentials and fitting of MTRM(T) and τ(T).(a) The variation of a stretched exponential, M = M0 exp[−(t/τ)β], for different values of β using a relaxation time τ = 1000 s. The black dots correspond to the recorded relaxation data for the d = 420 nm array at 200 K. In the inset the same data is fitted to a stretched exponential with β ≈ 0.15 and τ ≈ 1000 s, in the time window 10−10 (0.1 ns) to 1010 s (≈300 years) which covers the majority of the relaxation from M0 to zero magnetization. The magnetization is normalized to M0, which is the magnetization directly after the field is switched off (t = t0), corresponding to a fully dressed state of the array, see Fig. 1 (I). (b) A fit of the temperature dependence of the relaxation time τ to the Vogel Fulcher law, τ = τ0 exp(EB/kB(T − T0)), for both arrays, yielding an average energy barrier of EB/kB = 4500 K.
Mentions: The relaxation in Fig. 3 occurs across a large time window and it is not described by a simple exponential function. However, a stretched exponential, M = M0 exp[−(t/τ)β], where β is the stretching exponent, M0 is the magnetization at t = 0 and τ is the relaxation time, can be used to describe the relaxation process1318. To illustrate the influence of β, the stretched exponential decay in the time window 0.1 to 10 000 s is plotted in Fig. 5a, using τ = 1000 s and M0 = 1. As can be seen in Fig. 5a, a simple exponential (β = 1) decays rapidly and does not describe the relaxation behavior illustrated in Fig. 3. In Fig. 5a the relaxation data measured at 200 K is shown for the 420 nm array; comparing this to the stretched exponential curves one observes that it accurately matches the β = 0.15 curve. The 200 K data was also fitted to a stretched exponential decay, yielding τ ≈ 1000 s and β ≈ 0.15; red line in the inset of Fig. 5a. Extending this fitting procedure to the measured relaxation data, presented in Fig. 3a,b, the temperature evolution of τ can be studied. It should be noted that τ and β are strongly coupled and therefore physical meaningful fits can only be made in a relatively narrow temperature region (see supplementary material for more details). For a region of about 30 K, β is relative stable (around 0.15 for both arrays) and a reliable determination of the temperature dependence of τ can be made. For example, τ changes by 3 orders of magnitude from about 105 to 102 s for both arrays, in the interval 180 to 210 K for the d = 420 nm array and 200 to 230 K for the d = 380 nm array. The extracted τ(T) values were fitted to the Vogel-Fulcher law1920, τ = τ0exp[EB/kB(T − T0)], where τ0 = 10−11 s is a constant that describes the relaxation time of the individual islands at very high temperatures, T0 describes the interaction strength, EB is the energy barrier which the magnetic moment of the island has to overcome in order to reverse its direction and kB is the Boltzmann constant. The fits for both arrays are shown in Fig. 5b. From these fits, the energy barrier, EB, and the interaction strength T0, are derived. A value of EB/kB = 4650 K is received for the 380 nm array and 4430 K for the 420 nm array, while for T0 the values are 72 and 62 K, respectively. The results for T0 are consistent with a stronger interaction between the elements in the 380 nm array as compared to the elements in the 420 nm array, as expected. The energy barrier of the islands is expected to be rather similar for the two arrays, due to the fact that the islands have the same size and shape in the two arrays and the temperature ranges for the fits partly overlap. Therefore fits where the energy barrier was fixed to 4500 K were also made, yielding T0 = 77 K for the 380 nm array and T0 = 60 K for the 420 nm array. The individual blocking temperature, TB, of an island can be determined from τobs = τ0exp[EB/kBTB], where τobs is the observation time and is taken as 30 s. Using EB/kB  =  4500 K a blocking temperature of approximately 160 K can be estimated. The energy barrier, EB, of the individual islands can mainly be attributed to their shape anisotropy. Using this approximation the energy barrier can be estimated from EB(T) = μoMs(T)2ΔNV/2, where μo is the vacuum magnetic permeability, Ms(T) is the magnetization of an island at temperature T, ΔN is the differential demagnetizing factor calculated using the Osborn methodology21 (see Supplementary materials for details), and V is the volume of the magnetic island. By using this approach and the temperature dependence of M described in the Methods, a mean value of the energy barrier was determined to be EB/kB = 4900 K at 195 K and EB/kB = 4500 K at 215 K. By comparing these calculated values of EB with the values estimated from the relaxation experiments, it can be seen that they are in good agreement.

View Article: PubMed Central - PubMed

ABSTRACT

The properties of natural and artificial assemblies of interacting elements, ranging from Quarks to Galaxies, are at the heart of Physics. The collective response and dynamics of such assemblies are dictated by the intrinsic dynamical properties of the building blocks, the nature of their interactions and topological constraints. Here we report on the relaxation dynamics of the magnetization of artificial assemblies of mesoscopic spins. In our model nano-magnetic system - square artificial spin ice – we are able to control the geometrical arrangement and interaction strength between the magnetically interacting building blocks by means of nano-lithography. Using time resolved magnetometry we show that the relaxation process can be described using the Kohlrausch law and that the extracted temperature dependent relaxation times of the assemblies follow the Vogel-Fulcher law. The results provide insight into the relaxation dynamics of mesoscopic nano-magnetic model systems, with adjustable energy and time scales, and demonstrates that these can serve as an ideal playground for the studies of collective dynamics and relaxations.

No MeSH data available.


Related in: MedlinePlus