212
6 Magnetism and Conduction
6.4 Helical spin order: Consider a one dimensional spin chain with sizeable first
(J 1 ) and second (J 2 ) neighbour interactions. In the mean-field approximation, the
energy of the system is given by
E =−NS
2
J 1 cos(θ ) + J 2 cos(2θ)
In most cases the spins will align either parallel or anti-parallel, depending on the
sign of J 1 , but for certain ratios of J 1 /J 2 spin arrangements can be observed with noncollinear spin moments. Such spin configurations are supposed to play an important
role in the ferroelectric properties of magnetic materials. (a) Check that the energy
expression is identical to the energy of the Neél state (Eq. 6.73) when J 2 = 0 and
J 1 < 0. (b) Is there any possibility for a non-collinear alignment when J 2 > 0?
(c) Find the three values of θ for which the energy is minimized and classify them as
antiferromagnetic, ferromagnetic or non-collinear solutions. (d) Calculate the angle
between two neighbouring sites with J 2 =−0.3|J 1 | and J 1 =±1.
References
1. M.B. Robin, P. Day, Adv. Inorg. Chem. Radiochem. 10, 247 (1967)
2. C. Zener, Phys. Rev. 82(3), 403 (1951)
3. M.B. Salamon, M. Jaime, Rev. Mod. Phys. 73, 583 (2001)
4. J.J. Girerd, V. Papaefthymiou, K.K. Surerus, E. Münck, Pure Appl. Chem. 61, 805 (1989)
5. J.J. Borrás-Almenar, J.M. Clemente-Juan, E. Coronado, A.V. Palii, B.S. Tsukerblatt, in Magnetism: Molecules to Materials, ed. by J.S. Miller, M. Drillon (Wiley-VCH, 2001), pp. 155–210,
chap. 5
6. P.W. Anderson, H. Hasegawa, Phys. Rev. 100(2), 675 (1955)
7. K. Yosida, Theory of Magnetism (Springer, Berlin, 1996)
8. S. Blundell, Magnetism in Condensed Matter (Oxford University Press, Oxford, 2001)
9. W.J. Caspers, Spin Systems (World Scientific, Singapore, 1989)
10. J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer,
Berlin, 2006)
11. Z. Barandiarán, L. Seijo, J. Chem. Phys. 89, 5739 (1988)
12. K. Doll, M. Dolg, P. Fulde, H. Stoll, Phys. Rev. B 52, 4842 (1995)
13. B. Paulus, Phys. Rep. 428, 1 (2006)
14. V.I. Anisimov, J. Zaanen, O.K. Andersen, Phys. Rev. B 44(3), 943 (1991)
15. L. Hedin, Phys. Rev. 139, A796 (1965)
16. P. Fulde, H. Stoll, Found. Phys. 30, 2049 (2000)
17. A. Stoyanova, C. Sousa, C. de Graaf, R. Broer, Int. J. Quantum Chem. 106, 2444 (2006)
18. C. Pisani, M. Schütz, S. Casassa, D. Usvyat, L. Maschio, M. Lorenz, A. Erba, Phys. Chem.
Chem. Phys. 14, 7615 (2012)
19. E. Koch, in Correlated Electrons: From Models to Materials, ed. by E. Pavarini, E. Koch,
M. Jarrell (Forschungszentrum Jülich GmbH Institute for Advanced Simulation, Jülich, 2012),
chap. 7
20. E. Kaxiras, Atomic and Electronic Structure of Solids (Cambridge University Press, Cambridge,
2003)
6 Magnetism and Conduction
6.4 Helical spin order: Consider a one dimensional spin chain with sizeable first
(J 1 ) and second (J 2 ) neighbour interactions. In the mean-field approximation, the
energy of the system is given by
E =−NS
2
J 1 cos(θ ) + J 2 cos(2θ)
In most cases the spins will align either parallel or anti-parallel, depending on the
sign of J 1 , but for certain ratios of J 1 /J 2 spin arrangements can be observed with noncollinear spin moments. Such spin configurations are supposed to play an important
role in the ferroelectric properties of magnetic materials. (a) Check that the energy
expression is identical to the energy of the Neél state (Eq. 6.73) when J 2 = 0 and
J 1 < 0. (b) Is there any possibility for a non-collinear alignment when J 2 > 0?
(c) Find the three values of θ for which the energy is minimized and classify them as
antiferromagnetic, ferromagnetic or non-collinear solutions. (d) Calculate the angle
between two neighbouring sites with J 2 =−0.3|J 1 | and J 1 =±1.
References
1. M.B. Robin, P. Day, Adv. Inorg. Chem. Radiochem. 10, 247 (1967)
2. C. Zener, Phys. Rev. 82(3), 403 (1951)
3. M.B. Salamon, M. Jaime, Rev. Mod. Phys. 73, 583 (2001)
4. J.J. Girerd, V. Papaefthymiou, K.K. Surerus, E. Münck, Pure Appl. Chem. 61, 805 (1989)
5. J.J. Borrás-Almenar, J.M. Clemente-Juan, E. Coronado, A.V. Palii, B.S. Tsukerblatt, in Magnetism: Molecules to Materials, ed. by J.S. Miller, M. Drillon (Wiley-VCH, 2001), pp. 155–210,
chap. 5
6. P.W. Anderson, H. Hasegawa, Phys. Rev. 100(2), 675 (1955)
7. K. Yosida, Theory of Magnetism (Springer, Berlin, 1996)
8. S. Blundell, Magnetism in Condensed Matter (Oxford University Press, Oxford, 2001)
9. W.J. Caspers, Spin Systems (World Scientific, Singapore, 1989)
10. J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer,
Berlin, 2006)
11. Z. Barandiarán, L. Seijo, J. Chem. Phys. 89, 5739 (1988)
12. K. Doll, M. Dolg, P. Fulde, H. Stoll, Phys. Rev. B 52, 4842 (1995)
13. B. Paulus, Phys. Rep. 428, 1 (2006)
14. V.I. Anisimov, J. Zaanen, O.K. Andersen, Phys. Rev. B 44(3), 943 (1991)
15. L. Hedin, Phys. Rev. 139, A796 (1965)
16. P. Fulde, H. Stoll, Found. Phys. 30, 2049 (2000)
17. A. Stoyanova, C. Sousa, C. de Graaf, R. Broer, Int. J. Quantum Chem. 106, 2444 (2006)
18. C. Pisani, M. Schütz, S. Casassa, D. Usvyat, L. Maschio, M. Lorenz, A. Erba, Phys. Chem.
Chem. Phys. 14, 7615 (2012)
19. E. Koch, in Correlated Electrons: From Models to Materials, ed. by E. Pavarini, E. Koch,
M. Jarrell (Forschungszentrum Jülich GmbH Institute for Advanced Simulation, Jülich, 2012),
chap. 7
20. E. Kaxiras, Atomic and Electronic Structure of Solids (Cambridge University Press, Cambridge,
2003)