- Email: [email protected]
Temperature dependence of RKKY interaction
Journal of Magnetism and Magnetic Materials 393 (2015) 331–333
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Temperature dependence of RKKY interaction A. Kundu n, S. Zhang Department of Physics, University of Arizona, Tucson 85721, AZ, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 30 January 2015 Accepted 3 May 2015
The temperature dependence of RKKY coupling is shown to decrease much slower than the conventional zero-temperature RKKY coupling for a certain range of the distance between two magnetic ions. The magnitude of the coupling, but not the sign, can change significantly as the temperature arises. We discuss implication of such temperature dependence on laser-induced magnetization dynamics and predict a temperature dependence of the interlayer coupling in magnetic multilayers. & 2015 Elsevier B.V. All rights reserved.
Keywords: RKKY interaction spintronics Available online 4 June 2015
1. Introduction The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, E = − I (R) S1·S2, describes magnetic coupling between two magnetic ions mediated by itinerant electrons, where Si (i¼1, 2) are spins of the ions, R is the distance between the ions, and the coupling strength is
I (R) = − A ∑ kk′
f k − fk ′
ϵk − ϵk
dynamics does not take into account the possible temperature dependence of the coupling mediated by hot electrons. We show that the temperature dependence is much stronger than what one would expect from a simple correction due to temperature dependent density of states which is of the order of (k B T /ϵ F )2. In this short paper, we explicitly calculate the temperature dependence from Eq. (1) and we discuss several consequences of our results.
ei (k′− k)·R
′
(1)
where A is a constant and fk is a Fermi distribution function. For
2. Methods
the free electron model where ϵ k = = 2k 2/2m, one may analytically
We use the free electron model for the energy dispersion and
integrate out k and k′. At zero temperature where the Fermi distribution is a step function fk = θ (ϵ F − ϵ k ), the RKKY coupling is
fk = 1/ 1 + e β (ϵk −ϵ F ) . Since fk depends only on the magnitude of k,
I (R) ∝ (sin (2x ) − 2x cos (2x )) /x 4 where x = k F R , ϵF is the Fermi energy and kF is the Fermi wave number. At a large x, I(R) decays as R−3. The RKKY interaction is considered important for all conducting materials with local magnetic moments. Particularly, for rareearth and transition compounds, such as GdCoFe, itinerant conduction electrons provide dominant roles in magnetic coupling. Since the Fermi temperature of conduction electrons is typically two orders of magnitude higher than the room temperature, one can expect that the RKKY interaction is weakly temperature dependent. Recent experiments on laser-induced magnetization dynamics [1], however, involve electron temperature more than a thousand Kelvin although the spin temperature remains to be much lower [2]. Up till now, the analysis of these laser induced n
Corresponding author. E-mail address: [email protected] (A. Kundu).
http://dx.doi.org/10.1016/j.jmmm.2015.05.094 0304-8853/& 2015 Elsevier B.V. All rights reserved.
(
)
one can immediately integrate out the directions of k and k′,
I (R) =
C (k F R)2
∫0
∞
dk k sin (kR) fk
∫0
∞
dk ′
k′ sin (k′R) k′ 2 − k 2
(2)
where C = − (mV 2k F2/= 2π 4 ) A and V is the total volume of the system. The integration over k′ can easily be carried out by the contour integration, namely,
∞
(k′R ) = π cos (kR ). Thus, the ∫0 dk′ k′ sin k 2 − k2
′
RKKY coupling reduces to just one integration,
I (R) =
πC 2 (k F R)2
∫0
∞
dk k sin (2kR) fk .
(3)
For an arbitrary temperature, the above integration needed to be carried out numerically. We should first consider the case where k B T ≪ ϵ F . A frequently used approximation is the Sommerfeld expansion (SE) where
332
∫0
∞
A. Kundu, S. Zhang / Journal of Magnetism and Magnetic Materials 393 (2015) 331–333
dϵ g (ϵ) f (ϵ) =
∫0
ϵF
dϵ g (ϵ) +
π2 dg (ϵ) (k B T )2 dϵ 6
4
.
3
As long as the function g (ϵ) is a slowly varying function within the range of k B T near the Fermi level, SE gives a good approximation. By applying above SE to Eq. (3), we arrive at
(5)
where B = (mV 2k F4/8π 3= 2) A . The first term within the bracket above represents the coupling at zero temperature, which decays as 1/R 3 as noted before. Interestingly, the temperature dependent coupling, the second term in Eq. (5), decays as 1/x , much slower than the first term. Although (k B T /ϵ F )2 is usually small for metals even at a temperature as large as a thousand Kelvin degree, the second term can be comparable to the first term when the separation between magnetic ions is large. One may understand the different range dependence of the coupling as follows. The wave length of spin density waves of conduction electrons is 2π /k and thus these waves interfere destructively for different wave numbers when the waves propagate from one magnetic ion to another, see Eq. (3) which contains integration over sin (2kR ). When the sum over the entire Fermi sea is carried out, the net contribution to the coupling comes from the electrons near the Fermi surface. Quantitatively, at zero temperature the number of electrons gives arise to the RKKY coupling would be those within δk = π /2R of the Fermi wave number kF – which is explicitly shown in Eq. (3) if one replaces fk by a step function. For the temperature dependence of the RKKY interaction, the energy of electrons is narrowly distributed, i.e., ϵ F − k B T < ϵk < ϵ F + k B T . If the temperature is small, all excited states contribute to the coupling. This observation is in fact similar to the dynamic RKKY interaction where the frequency of the AC susceptibility can be viewed as a broadening of the Fermi level by =ω [5,6].
3. Results The slower decay of the temperature dependence of RKKY interaction, Eq. (5), however, is no longer valid if k B T > ϵ F /2k F R . In this case, sin (2kR ) cannot be treated as a slowly varying function within the energy range of kBT and thus the Sommerfeld expansion
Temperature dependent RKKY coupling
2
−400 −600 −800 2 4 6 8 10
1 0 −1 −2 −3
3000K
−4
1500K 750K 0K
−5 15
20
25
30
35
40
Distance k R F
Fig. 2. The RKKY interaction as a function of distance x = k F R for several temperatures. The inset shows the RKKY coupling at small values of x which has negligible temperature dependence. We take ϵ F = 3.0 eV .
fails. In Fig. 1, we show the results from Eq. (5) and the numerical results from Eq. (3) for the same set of parameters. As expected, the deviation becomes significant when the temperature becomes high or when the distance between two impurities is large. In Fig. 2, we calculate the RKKY coupling for the broad spectrum of the temperature and the distance. Since the Sommerfeld approximation fails at a large range and high temperatures, we use the exact numerical calculation directly from Eq. (3). Several interesting features are immediately recognized: (1) the temperature dependence is very weak for a small k F R , (2) the sign of the RKKY coupling does not change with temperature, but the magnitude decreases when the temperature increases, (3) the temperature dependence becomes significant for a large k F R > 20. For example, the RKKY coupling is reduced by 70% of its low temperature value at T = 1500 K and k F R ≃ 22, even though the thermal energy is less than 10% of the Fermi energy; in this case, a naive estimation, based on the scaling of temperature relative to the Fermi temperature, would be only about [I (T ) − I (0)]/ I (0) ≈ (k B T /ϵ F )2 ≈ 1% .
Temperature=1500K Sommerfeld expansion
1.5
4. Discussion
Exact result
1 0.5 0 −0.5 −1 −1.5 −2 0
−200
2
RKKY Coupling
⎡ ⎤ 2 π 2 ⎛ k B T ⎞ cos (2x) ⎥ sin (2x) − 2x cos (2x) + I (R) = − B ⎢ ⎜ ⎟ 4 ⎢⎣ ⎥⎦ x 3 ⎝ ϵF ⎠ x
180 0
(4)
ϵ=ϵ F
5
10
15
20
25
Distance k R F
Fig. 1. The temperature dependence of the RKKY interaction I (T ) − I (T = 0) as a function of k F R ), (a) calculated from the Sommerfeld expansion of Eq. (4) and (b) calculated by numerical integration of Eq. (3). We take ϵ F = 3.0 eV .
Finally, we discuss a possible application of our result on laser induced magnetization dynamics [1]. When a strong laser beam is applied to a magnetic film, the electron temperature could be raised to several thousand degrees of Kelvin [2]. The hot electrons transfer their energy to magnetic degrees of freedom, causing the demagnetization of the long-range magnetic ordering. The physical processes of the demagnetization involve highly non-equilibrium interactions and it has been shown that the exchange coupling at the picosecond scale takes different strengths for each magnetic element [3]. Several models, e.g., the atomistic simulation [4], have assumed that the exchange interaction between magnetic ions remains same in the presence of hot electrons. Our calculation indicates that the magnetic coupling is significantly reduced and the effective ordering temperature of the magnetic ions becomes much lower. Thus a smaller energy transfer from the hot electron to the magnetic ions can destroy the long range ordering, leading to faster magnetization dynamics.
A. Kundu, S. Zhang / Journal of Magnetism and Magnetic Materials 393 (2015) 331–333
A quantitative comparison between this calculation and experiments might be possible if a well-defined experimental system can be studied. We propose a magnetic multilayer such as Co/Cu and Fe/Cr where the interlayer magnetic coupling is known to be dominated by the RKKY interaction [7,8] and the thickness of the spacer layer Cu or Cr can be well-controlled. In early experiments [9,10], strong temperature dependence of exchange coupling in Co/Cu multilayers had been found in much lower temperatures (from 2 K to 300 K) whose origin of the temperature dependence was attributed to the granular structure of the interface, which is different from the intrinsic temperature dependence we have discussed here. If one is able to determine the coupling at high temperature for these multilayer systems at high temperature, either by laser heating or electric current heating, one would have an ideal case for theory-experiment comparison. In summary, we have calculated the RKKY interaction as a function of temperature. It is found that the range dependence of the RKKY interaction differs for low and high temperatures. The temperature dependence could be much stronger than the simple energy scaling of (k F T /ϵ F )2 would suggest.
Acknowledgements This work is supported by NSF-ECCS-1044542.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Andrei Kirilyuk, A.V. Kimel, Theo Rasing, Rev. Mod. Phys. 82 (2010) 2731. I. Radu, et al., Nature 472 (2011) 205–208. S. Mathias, et al., Proc. Natl. Acad. Sci. USA 109 (2012) 4792. T.A. Ostler, et al., Nat. Commn. 3 (2012) 666. G.M. Genkin, J. Appl. Phys. 81 (1997) 5310. S.R. Power, et al., Phys. Rev. B 85 (2012) 195411. S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. P. Bruno, C. Chappert, Phys. Rev. Lett. 67 (1991) 1602. N. Persat, A. Dinia, Phys. Rev. B 56 (1997) 2676. C. Christides, J. Appl. Phys. 88 (2000) 3552.
333
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Temperature dependence of RKKY interaction A. Kundu n, S. Zhang Department of Physics, University of Arizona, Tucson 85721, AZ, United States
art ic l e i nf o
a b s t r a c t
Article history: Received 30 January 2015 Accepted 3 May 2015
The temperature dependence of RKKY coupling is shown to decrease much slower than the conventional zero-temperature RKKY coupling for a certain range of the distance between two magnetic ions. The magnitude of the coupling, but not the sign, can change significantly as the temperature arises. We discuss implication of such temperature dependence on laser-induced magnetization dynamics and predict a temperature dependence of the interlayer coupling in magnetic multilayers. & 2015 Elsevier B.V. All rights reserved.
Keywords: RKKY interaction spintronics Available online 4 June 2015
1. Introduction The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, E = − I (R) S1·S2, describes magnetic coupling between two magnetic ions mediated by itinerant electrons, where Si (i¼1, 2) are spins of the ions, R is the distance between the ions, and the coupling strength is
I (R) = − A ∑ kk′
f k − fk ′
ϵk − ϵk
dynamics does not take into account the possible temperature dependence of the coupling mediated by hot electrons. We show that the temperature dependence is much stronger than what one would expect from a simple correction due to temperature dependent density of states which is of the order of (k B T /ϵ F )2. In this short paper, we explicitly calculate the temperature dependence from Eq. (1) and we discuss several consequences of our results.
ei (k′− k)·R
′
(1)
where A is a constant and fk is a Fermi distribution function. For
2. Methods
the free electron model where ϵ k = = 2k 2/2m, one may analytically
We use the free electron model for the energy dispersion and
integrate out k and k′. At zero temperature where the Fermi distribution is a step function fk = θ (ϵ F − ϵ k ), the RKKY coupling is
fk = 1/ 1 + e β (ϵk −ϵ F ) . Since fk depends only on the magnitude of k,
I (R) ∝ (sin (2x ) − 2x cos (2x )) /x 4 where x = k F R , ϵF is the Fermi energy and kF is the Fermi wave number. At a large x, I(R) decays as R−3. The RKKY interaction is considered important for all conducting materials with local magnetic moments. Particularly, for rareearth and transition compounds, such as GdCoFe, itinerant conduction electrons provide dominant roles in magnetic coupling. Since the Fermi temperature of conduction electrons is typically two orders of magnitude higher than the room temperature, one can expect that the RKKY interaction is weakly temperature dependent. Recent experiments on laser-induced magnetization dynamics [1], however, involve electron temperature more than a thousand Kelvin although the spin temperature remains to be much lower [2]. Up till now, the analysis of these laser induced n
Corresponding author. E-mail address: [email protected] (A. Kundu).
http://dx.doi.org/10.1016/j.jmmm.2015.05.094 0304-8853/& 2015 Elsevier B.V. All rights reserved.
(
)
one can immediately integrate out the directions of k and k′,
I (R) =
C (k F R)2
∫0
∞
dk k sin (kR) fk
∫0
∞
dk ′
k′ sin (k′R) k′ 2 − k 2
(2)
where C = − (mV 2k F2/= 2π 4 ) A and V is the total volume of the system. The integration over k′ can easily be carried out by the contour integration, namely,
∞
(k′R ) = π cos (kR ). Thus, the ∫0 dk′ k′ sin k 2 − k2
′
RKKY coupling reduces to just one integration,
I (R) =
πC 2 (k F R)2
∫0
∞
dk k sin (2kR) fk .
(3)
For an arbitrary temperature, the above integration needed to be carried out numerically. We should first consider the case where k B T ≪ ϵ F . A frequently used approximation is the Sommerfeld expansion (SE) where
332
∫0
∞
A. Kundu, S. Zhang / Journal of Magnetism and Magnetic Materials 393 (2015) 331–333
dϵ g (ϵ) f (ϵ) =
∫0
ϵF
dϵ g (ϵ) +
π2 dg (ϵ) (k B T )2 dϵ 6
4
.
3
As long as the function g (ϵ) is a slowly varying function within the range of k B T near the Fermi level, SE gives a good approximation. By applying above SE to Eq. (3), we arrive at
(5)
where B = (mV 2k F4/8π 3= 2) A . The first term within the bracket above represents the coupling at zero temperature, which decays as 1/R 3 as noted before. Interestingly, the temperature dependent coupling, the second term in Eq. (5), decays as 1/x , much slower than the first term. Although (k B T /ϵ F )2 is usually small for metals even at a temperature as large as a thousand Kelvin degree, the second term can be comparable to the first term when the separation between magnetic ions is large. One may understand the different range dependence of the coupling as follows. The wave length of spin density waves of conduction electrons is 2π /k and thus these waves interfere destructively for different wave numbers when the waves propagate from one magnetic ion to another, see Eq. (3) which contains integration over sin (2kR ). When the sum over the entire Fermi sea is carried out, the net contribution to the coupling comes from the electrons near the Fermi surface. Quantitatively, at zero temperature the number of electrons gives arise to the RKKY coupling would be those within δk = π /2R of the Fermi wave number kF – which is explicitly shown in Eq. (3) if one replaces fk by a step function. For the temperature dependence of the RKKY interaction, the energy of electrons is narrowly distributed, i.e., ϵ F − k B T < ϵk < ϵ F + k B T . If the temperature is small, all excited states contribute to the coupling. This observation is in fact similar to the dynamic RKKY interaction where the frequency of the AC susceptibility can be viewed as a broadening of the Fermi level by =ω [5,6].
3. Results The slower decay of the temperature dependence of RKKY interaction, Eq. (5), however, is no longer valid if k B T > ϵ F /2k F R . In this case, sin (2kR ) cannot be treated as a slowly varying function within the energy range of kBT and thus the Sommerfeld expansion
Temperature dependent RKKY coupling
2
−400 −600 −800 2 4 6 8 10
1 0 −1 −2 −3
3000K
−4
1500K 750K 0K
−5 15
20
25
30
35
40
Distance k R F
Fig. 2. The RKKY interaction as a function of distance x = k F R for several temperatures. The inset shows the RKKY coupling at small values of x which has negligible temperature dependence. We take ϵ F = 3.0 eV .
fails. In Fig. 1, we show the results from Eq. (5) and the numerical results from Eq. (3) for the same set of parameters. As expected, the deviation becomes significant when the temperature becomes high or when the distance between two impurities is large. In Fig. 2, we calculate the RKKY coupling for the broad spectrum of the temperature and the distance. Since the Sommerfeld approximation fails at a large range and high temperatures, we use the exact numerical calculation directly from Eq. (3). Several interesting features are immediately recognized: (1) the temperature dependence is very weak for a small k F R , (2) the sign of the RKKY coupling does not change with temperature, but the magnitude decreases when the temperature increases, (3) the temperature dependence becomes significant for a large k F R > 20. For example, the RKKY coupling is reduced by 70% of its low temperature value at T = 1500 K and k F R ≃ 22, even though the thermal energy is less than 10% of the Fermi energy; in this case, a naive estimation, based on the scaling of temperature relative to the Fermi temperature, would be only about [I (T ) − I (0)]/ I (0) ≈ (k B T /ϵ F )2 ≈ 1% .
Temperature=1500K Sommerfeld expansion
1.5
4. Discussion
Exact result
1 0.5 0 −0.5 −1 −1.5 −2 0
−200
2
RKKY Coupling
⎡ ⎤ 2 π 2 ⎛ k B T ⎞ cos (2x) ⎥ sin (2x) − 2x cos (2x) + I (R) = − B ⎢ ⎜ ⎟ 4 ⎢⎣ ⎥⎦ x 3 ⎝ ϵF ⎠ x
180 0
(4)
ϵ=ϵ F
5
10
15
20
25
Distance k R F
Fig. 1. The temperature dependence of the RKKY interaction I (T ) − I (T = 0) as a function of k F R ), (a) calculated from the Sommerfeld expansion of Eq. (4) and (b) calculated by numerical integration of Eq. (3). We take ϵ F = 3.0 eV .
Finally, we discuss a possible application of our result on laser induced magnetization dynamics [1]. When a strong laser beam is applied to a magnetic film, the electron temperature could be raised to several thousand degrees of Kelvin [2]. The hot electrons transfer their energy to magnetic degrees of freedom, causing the demagnetization of the long-range magnetic ordering. The physical processes of the demagnetization involve highly non-equilibrium interactions and it has been shown that the exchange coupling at the picosecond scale takes different strengths for each magnetic element [3]. Several models, e.g., the atomistic simulation [4], have assumed that the exchange interaction between magnetic ions remains same in the presence of hot electrons. Our calculation indicates that the magnetic coupling is significantly reduced and the effective ordering temperature of the magnetic ions becomes much lower. Thus a smaller energy transfer from the hot electron to the magnetic ions can destroy the long range ordering, leading to faster magnetization dynamics.
A. Kundu, S. Zhang / Journal of Magnetism and Magnetic Materials 393 (2015) 331–333
A quantitative comparison between this calculation and experiments might be possible if a well-defined experimental system can be studied. We propose a magnetic multilayer such as Co/Cu and Fe/Cr where the interlayer magnetic coupling is known to be dominated by the RKKY interaction [7,8] and the thickness of the spacer layer Cu or Cr can be well-controlled. In early experiments [9,10], strong temperature dependence of exchange coupling in Co/Cu multilayers had been found in much lower temperatures (from 2 K to 300 K) whose origin of the temperature dependence was attributed to the granular structure of the interface, which is different from the intrinsic temperature dependence we have discussed here. If one is able to determine the coupling at high temperature for these multilayer systems at high temperature, either by laser heating or electric current heating, one would have an ideal case for theory-experiment comparison. In summary, we have calculated the RKKY interaction as a function of temperature. It is found that the range dependence of the RKKY interaction differs for low and high temperatures. The temperature dependence could be much stronger than the simple energy scaling of (k F T /ϵ F )2 would suggest.
Acknowledgements This work is supported by NSF-ECCS-1044542.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Andrei Kirilyuk, A.V. Kimel, Theo Rasing, Rev. Mod. Phys. 82 (2010) 2731. I. Radu, et al., Nature 472 (2011) 205–208. S. Mathias, et al., Proc. Natl. Acad. Sci. USA 109 (2012) 4792. T.A. Ostler, et al., Nat. Commn. 3 (2012) 666. G.M. Genkin, J. Appl. Phys. 81 (1997) 5310. S.R. Power, et al., Phys. Rev. B 85 (2012) 195411. S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. P. Bruno, C. Chappert, Phys. Rev. Lett. 67 (1991) 1602. N. Persat, A. Dinia, Phys. Rev. B 56 (1997) 2676. C. Christides, J. Appl. Phys. 88 (2000) 3552.
333