ISSN 00213640, JETP Letters, 2011, Vol. 93, No. 7, pp. 384–387. © Pleiades Publishing, Inc., 2011. Original Russian Text © I.A. Ryzhkin, M.I. Ryzhkin, 2011, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 93, No. 7, pp. 426–430.
Screening of the Magnetic Field by Magnetic Monopoles in Spin Ice I. A. Ryzhkin* and M. I. Ryzhkin Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow region, 141701 Russia *email:
[email protected] Received February 2, 2011; in final form, February 18, 2011
The screening of the external magnetic field by magnetic monopoles in spin ice has been considered. The polarization of the magnetic system with moving monopoles has been shown to result in the incomplete screening of the external magnetic field. The static permeability of spin ice and the magneticfield screening length have been calculated and numerically estimated and the physical meaning of introducing monopoles is discussed. DOI: 10.1134/S0021364011070095
The concept of spin ice was introduced by Harris et al. [1] for designating Dy2Ti2O7 , Ho2Ti2O7 , and Yb2Ti2O7type materials. The name originates from the analogy between the distribution of spin orienta tions in the magnetic materials and the distribution of the dipole moments of hydrogen bonds in usual ice [2]. It turned out that the Hamiltonians of real spins in spin ice and the pseudospin Hamiltonian of usual ice formally have the same form of an antiferromagnetic Ising Hamiltonian with spins at the vertices of bound regular tetrahedra shown in Fig. 1. The tetrahedra themselves form a zinc blende lat tice; the same lattice is formed by ions in II–VI semi conductors or oxygen ions in cubic ice Ic. Owing to the presence of closed cycles with an odd number of verti ces, all antiferromagnetic ordering rules cannot be simultaneously fulfilled. Such lattices are referred to as frustrated ones and the corresponding materials are called geometrically frustrated magnets. The frustra tion leads to the macroscopic degeneracy of the ground state and the presence of nonzero entropy at T 0. Spin orientations in the lowest energy states must satisfy the following ice rule: two spins of each tetrahe dron are directed toward the center and two others are directed outward. Spin orientation in the ground state is exemplified in Figs. 1 and 2; the spins directed toward (outward) the tetrahedron centers in Fig. 1 are shown by closed (open) circles. As follows from Fig. 2, the spin orientations in the ground state are frozen, because the reorientation of any spin leads to violation of the ice rule and to an increase in the energy. How ever, there can be two types of ice rule violations at finite temperatures, i.e., tetrahedra with three spins directed toward the tetrahedron center (plus one spin directed outwards) and those with three spins directed
outward (plus one spin directed toward the center), which are referred to as positive and negative magnetic defects, respectively [3]. The positive and negative defects are created in pairs, but can be further spatially separated without additional violation of the ice rule. The creation and spatial separation of a defect pair are illustrated in Figs. 3 and 4, respectively. Let us mention two important features of the ground state. First, the reorientation of each spin entering the defect results in the transfer of the defect to an adjacent vertex rather than in its annihilation (if there is no defect of the opposite sign or surface nearby). Second, the defects moving in the lattice reorient spins on their way. In fact, a pair of created and then spatially separated
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Fig. 1. Zinc blende lattice formed by bound regular tetra hedra. Magnetic ions are at the vertices of the tetrahedra. The magnetic moments directed toward (outward) the tet rahedron centers are shown by closed (open) circles.
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defects remains a coupled string of spins aligned along the path of the defects. As was shown in our previous work [3], positive and negative defects can be regarded as quasiparticles with positive and negative magnetic charges, i.e., as mag netic monopoles. Later on, similar quasiparticles were predicted by Castelnovo et al. [4] from other consider ations and observed in experiments [5–8]. The latter works have brought great attention to the investigation of spin ice. In this context, the analysis of the experi ments in which magnetic monopoles could manifest themselves most clearly is of particular interest. In this work, we theoretically consider the problem of the screening of the magnetic field in spin ice and calcu late its static permeability. For convenience, we briefly recall the basic results of our previous work [3], which will be used below. Let us first consider a node with a positive magnetic defect (the vertex marked by the closed circle in Fig. 3) and surround the node by a closed surface containing only one tetrahedron. As is easily seen, the magnetization flux through the surface is nonzero. Consequently, the divergence of the magnetization is also nonzero, i.e., divM ≠ 0, which implies that the region occupied by the positive magnetic defect contains a positive mag netic charge. The magnetic field is determined by the equation similar to Poisson’s equation in electrostat ics, divH + 4πdivM = 0 ⇒ divH = 4πρ m ,
(1)
where ρm = –divM is the magnetic charge density introduced by the analogy with the electric charge density. The magnetic charge of the monopoles is qm± = ±2m/a, where m is the magnetic moment of magnetic atoms and a is the distance between the cen ters of the nearest tetrahedra. The force acting on the magnetic charge qm in the magnetic field H is f = qmH [3]. Let the halfspace x > 0 be occupied by spin ice and the homogeneous external magnetic field H0 be applied along the x axis. Under the action of the mag netic field, the monopoles are redistributed in space and screen the external magnetic field. The problem is to find the stationary distribution of the monopoles. A fundamental difference from the usual electrostatic problem is that the moving monopole orients the lat tice in such a way that the second one cannot pass along the same path, as is obviated by Fig. 4. Thus, the flux of monopoles leads to the polarization of the ground state, i.e., to the appearance of a vector field and to a decrease in the entropy of the system. A con venient way of taking these effects into account was elaborated in the physics of ice [9, 10]. This method for the problem at hand is as follows. Ordering due to the motion of monopoles can be described by a config JETP LETTERS
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Fig. 2. Fragment of the spin ice lattice with the magnetic ions represented by arrows. The directions of the magnetic moments in the ground state obey the ice rule: each vertex includes two incoming and two outgoing arrows.
Fig. 3. Creation of a pair of magnetic monopoles by the flip of the magnetic moment of the vertical bond: positive (negative) magnetic monopoles are shown by closed (open) circles.
Fig. 4. Migration of a positive magnetic monopole due to the flip of the magnetic moment of the vertical bond. The migration of the positive magnetic monopole orients mag netic moments along the migration trajectory.
uration vector, which is expressed in terms of the monopole flux densities as t
Ω ( r, t ) – Ω ( r, 0 ) =
∫ [ j ( r, t' ) – j ( r, t' ) ] dt', 1
0
2
(2)
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where the subscripts i = 1, 2 refer to the positive and negative monopoles, respectively. The monopole flux densities are in turn determined by the driving forces according to the linear expressions μi ni j i = ( q mi H – η i ΦΩ ) – D i ∇n i , (3) mi where μi, ni, qmi, and Di are the mobility, density, charge, and diffusion constant of monopoles, respec tively; η1, 2 = ±1; and Φ = 8akT/ 3 . Equation (3) shows that the expression qmiH – ηiΦΩ has a meaning of the driving force acting on the monopoles. In par ticular, the defect flux densities are nonzero in a partly ordered system with a nonzero configuration vector Ω ≠ 0, even in the absence of the external magnetic field and density gradient. For the stationary distribu tion of the defects, we set ji = 0; using this condition, we obtain the densities q mi ψ – η i Φχ⎞ , (4) n i = n 0 exp ⎛ – ⎝ ⎠ kT where the potentials ψ, χ are coupled to the field pro jections onto the x axis by the relations H = –dψ/dx, Ω = –dχ/dx, the defect density in the absence of fields is n0 = (2N/3V)exp(–⑀0/2kT), ⑀0 is the creation energy of a monopole pair, N/V = 3 3 /8a3 is the concentra tion of the aforementioned tetrahedra, and μi /Di = q mi /kT are the Einstein relations between the mobil ity and diffusion constants. The potentials can be found from the set of two equations 2
2
d ψ/dx = 4πq m ( δn 1 – δn 2 ), 2
(5)
2
(6) d χ/dx = ( δn 1 – δn 2 ). Equation (5) is Poisson’s equation for the magnetic potential and Eq. (6) is obtained by taking the diver gence of Eq. (2) with the use of the continuity equa tion. Combining Eqs. (5) and (6) and introducing the dimensionless function F = (qmψ – Φχ)/kT and the coordinate ξ = λx with the parameter λ specified as 2
⑀0 ⎞ 4 ⎛ 4πq m λ = + 1⎞ exp ⎛ – , 2⎝ Φ ⎠ ⎝ ⎠ 2kT a we come to the dimensionless equation 2
2
2
(7)
d F/dξ = sinh ( F ). (8) Equation (8) should be supplemented by the boundary conditions. At ξ ∞, the monopole densities must approach the equilibrium values. Thus, Eq. (4) leads to the condition F ( ξ = ∞ ) = 0. (9) To find the boundary conditions at ξ = 0, let us assume that the monopole densities are finite everywhere, which implies the absence of a surface monopole den sity. Thus, owing to Eqs. (5) and (6), we have the con ditions at the surface
∂χ ( 0 ) = 0, H ( 0 ) = – ∂ψ Ω ( 0 ) = – ( 0 ) = H 0 . (10) ∂x ∂x Equations (10) yield the following condition for the dimensionless function F: (11) dF ( 0 )/dξ = – q m H 0 /kTλ. Let us solve Eq. (8) with conditions (9) and (11). Integrating Eq. (8) and using the condition (9), we obtain 1 dF 2 ⎛ ⎞ = cosh ( F ) – 1, 2 ⎝ dξ⎠
(12)
q m H 0⎞ 2 (13) cosh ( F 0 ) = 1 ⎛ + 1, 2 ⎝ kTλ ⎠ where F0 = F(ξ = 0). Integrating once more, we find the solution in the form F /2
–ξ
F /2
e 0 + 1 + ( e 0 – 1 )e (14) F ( ξ ) = 2 ln . F /2 F /2 –ξ e 0 + 1 – ( e 0 – 1 )e Equations (5) and (14) yield the following relation for the magnetic field: 8πq m n 0 dH (15) = – sinh ( F ). dξ λ Formula (15) provides the numerical calculation of the magnetic field as a function of the distance from the surface. Simpler formulas appear in two limiting cases of F < F0 Ⰶ 1 and F0 > F Ⰷ 1. In the former case, expanding the exponentials to the linear terms, we find the solution in the form F(ξ) = F0e–ξ. Then, Eq. (15) yields 8πq m n 0 F 0 –ξ (16) H ( ξ ) = H 0 – ( 1 – e ), λ H0 (17) H ( ∞ ) = . 2 4πq m 1 + Φ Therefore, the steady state permeability of spin ice is 2
2 4πq 2 3πm . μ = 1 + m = 1 + 3 Φ a kT
(18)
As will be seen below, the linear approximation is valid only in the case of a weak external magnetic field on the order of several Gausses. In a strong field, using the condition F0 > F Ⰷ 1, we come to the formula for the coordinate dependence of the magnetic field F /2 ξ (19) H ( ξ ) = H 0 – ( 8πq m n 0 /λ )e 0 . – F /2 ξ + 2e 0 In this case, the field at infinity H(∞) is also given by Eq. (17). Thus, in a strong external field, the behavior of the magnetic field is initially (near the surface) described by Eq. (19), until the field becomes small enough to satisfy the lowfield approximation; we denote the corresponding value as H1. Deeper inside JETP LETTERS
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spin ice, the coordinate dependence of the field becomes exponential, similar to Eq. (16). Finally, deep in the sample, the field approaches a finite value spec ified by Eq. (17). This finite value of the magnetic field deep in the sample is the direct consequence of the aforementioned lattice polarization due to the motion of monopoles. In a similar problem of electrostatic screening in real ice, the electric field can be screened completely, since, in addition to ionic defects, there are also Bjerrum defects, whose motion polarizes the lattice in the opposite direction. Let us estimate the numerical values of various parameters and discuss the meaning of the results. We take rnn ≈ 3.54 × 10–8 cm for the distance between the adjacent spins [11]. Then, the distance between the centers of the adjacent tetrahedra is a ≈ 4.34 × 10–8 cm. The magnetic charge is qm = 2m/a ≈ 4.27 × 10–12 dyn/G. In the linear approximation, the permeability is given by the expression μ ≈ 1 + 8.29/T; i.e., the external magnetic field at temperature T = 1 K is weakened in spin ice by almost an order of magnitude. The parameter l = λ–1 has the meaning of the screening length for the weak magnetic field, whose screening is described by Eqs. (16) and (17). Accord ing to Eq. (7), ⑀0 ⎞ a (20) l = exp ⎛ . ⎝ 4kT⎠ 2 μ The screening length depends exponentially on the creation energy ⑀0 of the monopole pair. Both the exchange coupling constant J and the energy required for the separation of the monopoles to a large distance compared to the length a contribute to ⑀0. The cre ation energy must be calculated quite accurately owing to the exponential dependence in Eq. (20). It seems more reasonable to use Eq. (20) for the experi mental measurement of the creation energy of the monopole pair. Note that we can consider the mono poles as noninteracting quasiparticles only if the mag netostatic interaction between them is weak. For that, the equilibrium monopole density must be low; i.e., ⑀0/2kT Ⰷ 1. Let us take the value ⑀0 ≈ 25 K for illustra tion. Then, the screening length at temperature T = 1 K becomes l ≈ 3.69 × 10–6 cm. In this case, the monopole density is n0 ≈ 1.97 × 1016 cm–3 and the screening length coincides with the average distance – 1/3 n 0 ≈ 3.70 × 10–6 cm between the monopoles; i.e., we are just at the applicability limit of the approximation of continuous magnetic charge distribution.
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Let us estimate the applicability of the linear approximation F0 Ⰶ 1. Equation (12) yields the condi tion H0 Ⰶ kT/qml ≈ 8.75 G (at T = 1 K). Thus, the orderofmagnitude estimate of the previously intro duced magnetic field is H1 ≈ kT/qml. The linear approximation holds only in very low magnetic fields, whereas in higher fields one has to use more general Eqs. (12)–(14) or Eq. (19). The screening length in a high magnetic field is larger than the lowfield screen ing length l = λ–1. To conclude, the description of the screening of the external magnetic field in terms of monopoles is for mally equivalent to the description in terms of ele mentary dipoles. The difference is the complexity of the description. The orientations of the elementary dipoles are strongly correlated due to the ice rule and one has to deal with a strongly correlated manybody system. The violations of the rule, i.e., the monopoles, are much fewer and, under this condition, they can be treated as noninteracting quasiparticles. Such an approach (considering noninteracting excitations instead of original interacting particles) is common in the theory of strongly correlated manybody systems. For this reason, the formulation of the problem in terms of monopoles is simpler than that in terms of elementary dipoles. REFERENCES 1. M. J. Harris, S. T. Bramwell, D. F. McMorrow, et al., Phys. Rev. Lett. 79, 2554 (1997). 2. I. A. Ryzhkin, Solid State Commun. 52, 49 (1984). 3. I. A. Ryzhkin, Zh. Eksp. Teor. Fiz. 128, 559 (2005) [J. Exp. Theor. Phys. 101, 481 (2005)]. 4. C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature 451, 42 (2007). 5. S. T. Bramwell, S. R. Giblin, S. Calder, et al., Nature 461, 956 (2009). 6. T. Fennell, P. P. Deen, A. R. Wildes, et al., Science 326, 415 (2009). 7. D. J. Morris, D. A. Tennant, S. A. Grigera, et al., Sci ence 326, 411 (2009). 8. H. Kadowaki, N. Doi, Y. Aoki, et al., J. Phys. Soc. Jpn. 78, 103706 (2009). 9. C. Jaccard, Phys. Condens. Mater. 3, 99 (1964). 10. I. A. Ryzhkin and R. W. Whitworth, J. Phys.: Condens. Matter 9, 395 (1997). 11. R. Siddharthan, B. S. Shastry, A. P. Ramirez, et al., Phys. Rev. Lett. 83, 1854 (1999).
Translated by A. Safonov
