Electron Spin Resonance Shift in Spin Ladder Compounds

We analyze the effects of different coupling anisotropies in a spin-$1/2$ ladder on the electron spin resonance (ESR) shift. Combining a perturbative expression in the anisotropies with density matrix renormalization group computation of the short range correlations at finite temperature, we provide the full temperature and magnetic field evolution of the ESR paramagnetic shift. We show that for well chosen parameters the ESR shift can be in principle used to extract quantitatively the anisotropies and, as an example, discuss the material BPCB.

Figure 1

Magnetic field dependence of $Y_{\parallel }$ (lines with circles) for several temperatures and coupling ratios $j$. In (a)–(c), $Y_{\perp }$ is represented by lines without circles. In (d), the strong coupling $j\to \infty$ approximation (9) of $Y_{\parallel }$ is shown with lines without circles. For the BPCB coupling ratio $j=3.6$ in (c), the plotted temperatures are $T\approx 0.72\mathrm{K}$, $T\approx 1.44\mathrm{K}$, $T\approx 3.6\mathrm{K}$, $T\approx 10.8\mathrm{K}$ in experimental units. The two vertical dotted lines locate the critical fields $H_{c1}$ and $H_{c2}$ respectively.

Figure 2

Magnetic field dependence of (a) the ESR paramagnetic resonance frequency ${\omega }_r$, (b) the ESR resonance frequency shift $\delta \omega$. (c) Temperature dependence of the magnetic field at which the ESR paramagnetic resonance at ${\omega }_r=2\pi \times 96\mathrm{GHz}$ occurs. Symbols denote experimental data taken from [17] and lines theoretical predictions of the best fitting parameters $f_{\parallel 1}/k_B=0.1\mathrm{K}$, $f_{\perp 1}/k_B=-1.4\mathrm{K}$ and $f_{\parallel 2}/k_B=0.2\mathrm{K}$, $f_{\perp 2}/k_B=0.2\mathrm{K}$ if not stated otherwise. The results corresponding to the two differently oriented ladders 1 and 2 are plotted in black and red (gray), respectively. In (a) the linear Zeeman component of the ESR frequency resonance is represented by full lines (for the Landé factors $g_1$ and $g_2$). We show in (b) the theoretical prediction of the ESR shift at $T=1.3\mathrm{K}$ for the ladder 1 considering only the leg (black dotted lines) or the rung anisotropy (black dashed lines) with the fitting parameters (10, 11), respectively. These fits show only small deviations with the best fit in this range of temperature and magnetic field.

Figure 3

Theoretical prediction of the ESR shift $\delta \omega$ versus the magnetic field at $T=3\mathrm{K}$ for the two ladders’ orientations. The predictions are plotted in dotted, dashed, and full lines considering only the rung, leg anisotropy with the fitting parameters (10, 11), and with the best fitting parameters given in Fig. 2, respectively. The shaded area represents the standard deviation of the ESR experiments in Ref. 17 from our theoretical prediction. Note that for the experimental configuration of Ref. 17, the anisotropies along the magnetic field for ladder 1 could be well separated from the ESR measurements at $T=3\mathrm{K}$.