Field dependence of the magnetic spectrum in anisotropic and Dzyaloshinskii-Moriya antiferromagnets. II. Raman spectroscopy

We compare the theoretical predictions of the previous paper on the field dependence of the magnetic spectrum in anisotropic two-dimensional and Dzyaloshinskii-Moriya layered antiferromagnets [L. Benfatto and M. B. Silva Neto, Phys. Rev. B 74, 024415 (2006)], with Raman spectroscopy experiments in ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ and untwinned ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ single crystals. We start by discussing the crystal structure and constructing the magnetic point group for the magnetically ordered phase of the two compounds, ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ and ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$. We find that the magnetic point group in the ordered phase is the $\underset{̱}{m}m\underset{̱}{m}$ orthorhombic group, in both cases. Furthermore, we classify all the Raman active one-magnon excitations according to the irreducible co-representations for the associated magnetic point group. We find that the in-plane (or Dzyaloshinskii-Moriya) mode belongs to the $D{A}_g$ co-representation while the out-of-plane $\left(XY\right)$ mode belongs to the $D{B}_g$ co-representation. We then measure and fully characterize the evolution of the one-magnon Raman energies and intensities for low and moderate magnetic fields along the three crystallographic directions. In the case of ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$, a weak-ferromagnetic transition is observed for a magnetic field perpendicular to the $\mathrm{Cu}{\mathrm{O}}_2$ layers. We demonstrate that from the jump of the Dzyaloshinskii-Moriya gap at the critical magnetic field $H_c\simeq 6.6\mathrm{T}$ one can determine the value of the interlayer coupling $J_{\perp }∕J\simeq 3.2\times {10}^{-5}$, in good agreement with previous estimates. We furthermore determine the components of the anisotropic gyromagnetic tensor as $g_s^a=2.0$, $g_s^b=2.08$, and the upper bound $g_s^c=2.65$, also in very good agreement with earlier estimates from magnetic susceptibility. For the case of ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$, we compare the Raman data obtained in an in-plane magnetic field with previous magnon-gap measurements done by electron spin resonance (ESR). Using the very low magnon gap estimated by ESR $(\sim 0.05\mathrm{meV})$, the data for the one-magnon Raman energies agree reasonably well with the theoretical predictions for the case of a transverse field (only hardening of the gap). On the other hand, an independent fit of the Raman data provides an estimate for $g_s\simeq 1.98$ and gives a value for the in-plane gap larger than the one measured by ESR. Finally, because of the absence of the Dzyaloshinskii-Moriya interaction in ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$, no field-induced modes are observed for magnetic fields parallel to the $\mathrm{Cu}{\mathrm{O}}_2$ layers in the Raman geometries used, in contrast to the situation in ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$.

Figure 1

Tetragonal magnetic unit cell of ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ for $T<T_N$ (only ${\mathrm{Cu}}^{++}$, ${\mathrm{O}}^{--}$, and ${\mathrm{Cl}}^{-}$ ions are shown for clarity). For the sake of unifying notations between ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ and ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$, in what follows we shall use the $\left(abc\right)$ coordinate system above where $a\Vert x^{\prime }$, $b\Vert y^{\prime }$, and $c\Vert z$. Since the crystal structure is tetragonal, we have $a=b<c$. The ${\mathrm{Cu}}^{++}$ spins are confined to the $\left(ab\right)$ plane ($\mathrm{Cu}{\mathrm{O}}_2$ layers) and are oriented along the $\left(\overline{1}10\right)$ direction in the $\left(xyz\right)$ coordinate system, or, equivalently, parallel to the $b$ axis. In the paramagnetic phase, the point group is the $I4∕mmm$ tetragonal group, but, as we shall discuss soon, below $T_N$ the symmetry is lowered because of the antiferromagnetic ordering of the spins.

Figure 2

Left: filled (small) circles represent the ${\mathrm{Cu}}^{++}$ ions. The hatched larger circles represent the oxygen ${\mathrm{O}}^{--}$ ions that are tilted above the $\mathrm{Cu}{\mathrm{O}}_2$ layer while the open larger circles are ${\mathrm{O}}^{--}$ ions that are tilted below the $\mathrm{Cu}{\mathrm{O}}_2$ layer, see also Fig. 3. Right: the staggered pattern of the Dzyaloshinskii-Moriya vectors, represented by open arrows along the Cu-Cu bonds, follows from the staggered pattern of the tilting of the oxygen octahedra. Small arrow: in-plane projection of the ${\mathrm{Cu}}^{++}$ spins in the ordered phase.

Figure 3

Orthorhombic magnetic unit cell of ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ for $T<T_N$ (only ${\mathrm{Cu}}^{++}$ and ${\mathrm{O}}^{--}$ ions—except the apical ones—are shown for clarity). We use the $\left(abc\right)$ orthorhombic coordinate system, where $a\ne b\ne c$. The ${\mathrm{Cu}}^{++}$ ions order antiferromagnetically parallel to the $b$ orthorhombic axis but are also canted out of each $\mathrm{Cu}{\mathrm{O}}_2$ layer, and are thus confined to the $\left(bc\right)$ plane. Observe that the canting pattern is staggered along the $c$ orthorhombic direction.

Figure 4

(Color online) One-magnon Raman response for ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ at $T=10\mathrm{K}$ in the $\left(RL\right)$ circular polarization configuration, where only the Dzyaloshinskii-Moriya gap is Raman active. In (a) the gap only hardens, in (b) the gap only softens, while in (c) it first softens and then hardens above the weak-ferromagnetic transition at $H\simeq 6.6\mathrm{T}$ (see discussion in the text).

Figure 5

(Color online) One-magnon Raman response for ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ at $T=10\mathrm{K}$ in the $\left(RR\right)$ circular polarization configuration for a magnetic field applied along the orthorhombic $b$ easy axis, where the field-induced mode becomes Raman active (see discussion in the text). We observed a monotonic hardening of the field-induced mode with increasing applied field.

Figure 6

(Color online) One-magnon Raman response for ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ at $T=10\mathrm{K}$ in the $\left(RL\right)$ polarization configuration, where only the in-plane gap is Raman active. Main figure: hardening of the in-plane gap with increasing in-plane magnetic field. Insets: (top) zoomed image of the main plot, showing the much weaker intensity of the two-magnon scattering, starting at approximately twice the magnon gap; (bottom) featureless response for perpendicular magnetic field [the out-of-plane $\left(XY\right)$ mode is not Raman active in the backscattering geometry used here].

Figure 7

(Color online) Comparison between the experimental data and theoretical predictions for the field dependence of the DM gap for $\mathbf{H}\Vert a$. The closed circles are the experimental data. The line is the best fit obtained using Eq. (17), with ${\gamma }_a^F=0.93{\left(\mathrm{cm}\mathrm{T}\right)}^{-2}$.

Figure 8

(Color online) Phase diagram of the weak-ferromagnetic transition and spin configuration for $H\Vert c$. The arrows with open tip represent the spins in each layer, the small ones on the right of the layers the uniform spin components. At low temperature and zero or low field the uniform components of the spins in each layers are ordered antiferromagnetically in the $c$ direction. Above the critical field (19) the uniform components order ferromagnetically in the $c$ direction. As the temperature increases the uniform components decrease following the decrease of the AF spin component along $b$, and as a consequence also the critical field for the WF transition decreases. The phase diagram has been calculated in Ref. 1 (see Fig. 10) by using parameter values appropriate for ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$.

Figure 9

(Color online) Comparison between the experimental data and the theoretical predictions for the field dependence of the DM gap for $\mathbf{H}\Vert c$. The closed circles are the experimental data. Observe that near the WF transition two values of the gap have been measured [see also panel (c) of Fig. 4]. Indeed, since the transition is first order in a finite sample the transition is observed as a coexistence of the two phases (above and below the transition) for a range of field values around the thermodynamic critical field. The lines are the fit using Eqs. (24), with ${\gamma }_c^F=21.83{\mathrm{cm}}^{-2}{\mathrm{T}}^{-1}$.

Figure 10

(Color online) Phase diagram and evolution of the spin configuration for $H\Vert b$. For the spin components we use the same notation as in Fig. 8 above. At $H<H_c^{\left(1\right)}$ both the spins and the uniform components rotate in the $bc$ plane, so that the net uniform magnetization is along the $b$ axis. In this region $T_N\left(H\right)$ is the temperature at which the in-plane AF component along $b$ vanishes. At $H_c^{\left(1\right)}<H<H_c^{\left(2\right)}$ the AF spin components align along $a$ which is perpendicular to the plane of the figure, so that these spin components, which vanish at the temperature $T_N\left(H\right)$, are represented by the small crosses. At the same time, the uniform components align ferromagnetically in neighboring planes. At $H>H_c^{\left(2\right)}$ only the $c$ components survive.

Figure 11

(Color online) Comparison between the experimental data and the theoretical predictions for the field dependence of the $a$ mode for $\mathbf{H}\Vert b$. The solid line is the fit done using the first of Eqs. (34) with $g_s$ and $m_c$ as fitting parameters, giving $g_s^b=2.08$ and $m_c=36{\mathrm{cm}}^{-1}$. The dashed line is the approximated expression (35) evaluated with the same parameters values, and it is only valid at low field. Inset: field dependence of the $c$ mode observed at fields larger than $4\mathrm{T}$. The solid line is the curve corresponding to the second of Eqs. (34).

Figure 12

(Color online) Field dependence of the normalized intensity $I_a\left(H\right)∕I_a$ of the $a$ peak for a longitudinal field. We estimated the relative error on the measured peak intensity around 20%. Solid line: field dependence of the peak intensity according to Eqs. (39, 40). Observe that by neglecting the contribution of the $Z_a$ factor from Eq. (40) one would obtain $I_a=1∕{\omega }_a$, corresponding to the dashed line, which shows an increasing of the relative intensity not observed in the experiments.

Figure 13

(Color online) Field dependence of the intensity of the $a$ peak for a transverse field in the $c$ direction. Observe that the spin-flop WF transition is first order, so that by increasing the magnetic field one first accesses a crossover regime where the Raman intensity is split between the two peaks corresponding to the equilibrium gap values at $H<H_c$ and $H>H_c$. Thus only the data at $H>7\mathrm{T}$ should be compared with the dashed curve, corresponding to the state above the spin flop.

Figure 14

(Color online) Field dependence of the in-plane spin-wave gap measured by Raman in ${\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$. Solid line: field dependence of the $a$ gap obtained using the value $m_a=0.4{\mathrm{cm}}^{-1}$ and $g_s=2.05$ measured by ESR and the theoretical prediction for a transverse field. Dashed line: best fit on the data point using Eq. (43) with $m_a$ and $g_s$ as fitting parameters.