Uniaxial antiferromagnet

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 124-125

Section 1.5.3.3.2. Uniaxial antiferromagnet

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.3.3.2. Uniaxial antiferromagnet

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Now let us proceed to the uniaxial antiferromagnet with two ions in the primitive cell. The thermodynamic potential $[\tilde{\Phi}]$ for such an antiferromagnet is given in accordance with (1.5.3.26) and (1.5.3.27) by (Landau, 1933) $[\eqalignno{\tilde{\Phi} &= \Phi_0 + (A/2){\bf L}^2 + (B/2){\bf M}^2 + (a/2)(L_x^2 + L_y^2) &\cr&\quad+ (b/2)(M_x^2 + M_y^2) + (C/4){\bf L}^4 + (D/2)({\bf LM})^2 &\cr&\quad+ (D'/2){\bf L}^2{\bf M}^2 - {\bf MH}.&(1.5.3.37)} ]$

If the magnetic field is absent $[({\bf H}=0)]$ , then $[{\bf M} = 0]$ because B, D and [D'>0] . Then three possible magnetic states are obtained by minimizing the potential with respect to $[\bf L]$ only: $[\displaylines{{\rm (I)} \hfill L_x = L_y = L_z = 0 \hfill(1.5.3.38)\cr {\rm (II)} \hfill L_z = 0;\quad L_x^2 + L_y^2 = L_{\perp}^2 = -{{A + a}\over{C}} \hfill(1.5.3.39)\cr {\rm (III)} \hfill L_x = L_y = 0;\quad L_z^2 = -{{A}\over{C}}. \hfill(1.5.3.40)}%fd1.5.3.40]$

When $[a \,\lt\, 0]$ , state (II) with [L_z=0] is thermodynamically stable. When $[a\,\gt\,0]$ , state (III) is stable and the antiferromagnetic vector is directed along the axis. This means that the term with the coefficient a is responsible for the anisotropy of the uniaxial antiferromagnet. We introduce the effective anisotropy field: $[H_a = aL = 2aM_0, \eqno(1.5.3.41)]$ where [M_0] is the sublattice magnetization.

Formulas (1.5.3.39) and (1.5.3.14) in the form $[A=\lambda(T-T_c)]$ yield the expression for the temperature dependence of the sublattice magnetization: $[L^2 = (\lambda /C)(T_N - T), \eqno(1.5.3.42)]$ where [T_N] is the Néel temperature. The assertions relating to formulas (1.5.3.34) and (1.5.3.35) concerning the fluctuation region are also valid for the temperature dependence of the sublattice magnetization.

The minimization of the potential $[\tilde{\Phi}]$ with respect to $[\bf M]$ for given $[{\bf L} \neq 0]$ when $[{\bf H} \neq 0]$ yields the following relation for the magnetization: $[{\bf M} = \chi_{\perp}{\bf H} - (\chi_{\perp} - \chi_{ \parallel })({\bf qH}){\bf q}, \eqno(1.5.3.43)]$ where $[{\bf q} = {\bf L}/|L|]$ . Thus the magnetization of an antiferromagnet is linear with the magnetic field, as for a paramagnet, if the magnetic field is not too strong. The main difference is in the anisotropy and temperature dependence of the susceptibility. The parallel susceptibility $[{\chi}_{ \parallel }]$ decreases when the temperature is lowered, and $[{\chi}_{\perp}]$ does not depend on temperature ( $[\chi_{\perp} = 1/B]$ ) (see Fig. 1.5.3.6). The coefficient B belongs to the exchange term and defines the effective exchange field $[H_e = {\textstyle{1 \over 2}}BL = BM_0. \eqno(1.5.3.44)]$

Figure 1.5.3.6 | top | pdf |

Temperature dependence of the mass susceptibility $[\chi_{g}]$ for a uniaxial antiferromagnet along ( $[{\chi}_{ \parallel }]$ ) and perpendicular ( $[{\chi}_{\perp}]$ ) to the axis of antiferromagnetism (see Foner, 1963).

As seen from Fig. 1.5.3.6, $[\chi_{\perp}\,\gt\, \chi_{ \parallel }]$ . Therefore, when the magnetic field applied parallel to the axis of a uniaxial antiferromagnet reaches the critical value $[H_{c1}^2 = aL^2/(\chi_{\perp} - \chi_{ \parallel }) \simeq aBL_0^2 = 2H_aH_e \eqno(1.5.3.45)]$ ( [L_0] is the value of L at [T = 0] ), a flopping of the sublattices from the direction along the axis to some direction in the plane perpendicular to the axis occurs. In this spin-flop transition (which is a first-order transition into a new magnetic structure), the magnetization jumps as shown in Fig. 1.5.3.7.

Figure 1.5.3.7 | top | pdf |

Dependence of the relative magnetization $[M/M_{\rm max}]$ on the magnetic field at [T=0] . The dashed line corresponds to $[{\bf H} \perp Oz]$ , the full line to $[{\bf H} \parallel Oz]$ . $[H_{c1}]$ is the field of spin-flop, $[H_{c2}]$ is the field of spin-flip.

A second-order transition into a saturated paramagnetic state takes place in a much stronger magnetic field $[H_{c2} = 2H_e]$ . This transition is called a spin-flip transition. Fig. 1.5.3.7 shows the magnetic field dependence of the magnetization of a uniaxial antiferromagnet. Fig. 1.5.3.8 shows the temperature dependence of both critical fields.

Figure 1.5.3.8 | top | pdf |

Magnetic phase diagram for a uniaxial antiferromagnet in a magnetic field applied parallel to the axis. (1) The line of spin-flop transition $[(H_{c1})]$ ; (2) the line of spin-flip transition $[(H_{c2})]$ ; P, paramagnetic phase; AFM, easy-axis antiferromagnetic phase; SF, spin-flop phase; BP, bicritical point.

The quantitative behaviour of the critical magnetic fields in the neighbourhood of [T_N] for both directions of the magnetic field ( $[{\bf H} \parallel Oz]$ and $[{\bf H} \perp Oz]$ ) can be determined from the theory of second-order phase transitions starting from the thermodynamic potential $[\tilde{\Phi}]$ and taking into account that L is small and $[DL^2 \ll B]$ close to [T_N] .

In the presence of the magnetic field $[{\bf H} \perp Oz]$ , $[{\bf L}]$ is parallel to [Oz] , $[{\bf LM} = 0]$ , the coefficient A at [L^2] is replaced by [A + 2D'H^2/B^2] and the latter is zero at the new transition point. The critical field is given by the relation $[H_{c2}^2 = (\lambda B^2/2D')(T_N - T), \quad {\bf H} \perp Oz. \eqno(1.5.3.46)]$

If the field is applied parallel to the z axis, then $[{\bf L}]$ remains parallel to [Oz] if $[H \,\lt\, H_{c1}]$ ( $[H_{c1} \simeq aB^2/D]$ in the neighbourhood of [T_N] ). Therefore, $[H_{c2}^2 = {{{\lambda}B^2}\over{2(D + D')}}(T_N - T), \quad {\bf H} \parallel Oz,\;H \,\lt\, H_{c1}. \eqno(1.5.3.47)]$ If $[H> H_{c1}]$ , $[{\bf L}]$ becomes perpendicular to the z axis and the anisotropy term has to be taken into account: $[H_{c2}^2 = {{{\lambda}B^2}\over{2D'}}(T_N - T - a/{\lambda}), \quad {\bf H} \parallel Oz,\; H> H_{c1}. \eqno(1.5.3.48)]$

Formulas (1.5.3.46)–(1.5.3.48) show that the transition temperature is reduced by applying the magnetic field. The displacement of the transition point is directly proportional to the square of the applied field. Fig. 1.5.3.9 shows the phase diagram of an antiferromagnet in the neighbourhood of [T_N] . Unlike ferromagnets, antiferromagnets maintain the second-order phase transition when a magnetic field is applied because the symmetry of the crystal in the antiferromagnetic state differs essentially from that in the paramagnetic state also if the crystal is placed into a magnetic field.

Figure 1.5.3.9 | top | pdf |

Phase diagram for a uniaxial antiferromagnet in the proximity of [T_N] , calculated for MnCl₂·4H₂O. Experimental data are taken from Gijsman et al. (1959).

Formula (1.5.3.43) describes the magnetization process only in easy-axis antiferromagnets. For easy-plane antiferromagnets, the anisotropy in the plane is usually extremely small and the antiferromagnetic vector rotates freely in the basic plane. Therefore, for any direction of the magnetic field, the vector $[\bf L]$ becomes aligned perpendicular to the applied magnetic field. Correspondingly the magnetization becomes $[{\bf M} = {\chi}_zH_z\hat{\bf z} + {\chi}_{\perp}H_{\perp}\hat{\bf x}, \eqno(1.5.3.49)]$ where $[\hat{\bf z}]$ and $[\hat{\bf x}]$ are unit vectors parallel and perpendicular to the axis.

References

Landau, L. D. (1933). Eine mögliche Erklärung der Feldabhängigkeit der Suszeptibilität bei niedrigen Temperaturen. Phys. Z. Sowjet. 4, 675–679.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 124-125