Impurity effects in coupled-ladder ${\text{BiCu}}_2{\text{PO}}_6$ studied by NMR and quantum Monte Carlo simulations

We present a ${}^{31}\text{P}$ NMR study of the coupled spin $\frac{1}{2}$ ladder compound ${\text{BiCu}}_2{\text{PO}}_6$. In the pure material, intrinsic susceptibility, and dynamics show a spin gap of about $\Delta \approx 35–40\text{K}$. Substitution of nonmagnetic Zn or magnetic Ni impurity at Cu site induces a staggered magnetization which results in a broadening of the ${}^{31}\text{P}$ NMR line while susceptibility far from the defects is unaffected. The effect of Ni on the NMR line broadening is twice that of Zn, which is consistent with quantum Monte Carlo calculations assuming that Ni couples ferromagnetically to its adjacent Cu. The induced moment follows a $1/T$ temperature dependence due to the Curie-type development of the moment amplitude while its extension saturates and does not depend on impurity content or nature. This allow us to verify the generically expected scenario for impurity doping and to extend it to magnetic impurity case: in an antiferromagnetically correlated low-dimensional spin system with antiferromagnetic correlations, any type of impurity induces a staggered moment at low temperature, whose extension is not linked to the impurity nature but to the intrinsic physics at play in the undoped pure system, from one-dimensional to two-dimensional systems.

Figure 1

(Color online) Variation in the ${}^{31}\text{P}$ NMR spin shift $K_{spin}$ (left axis) with temperature $T$ is compared with that of the bulk susceptibility ${\chi }_{\text{bulk}}$ (right axis) for undoped ${\text{BiCu}}_2{\text{PO}}_6$. The inset zoom shows that $K_{spin}$ decreases and levels off at a small value at low $T$.

Figure 2

(Color online) The ${}^{31}\text{P}$ nuclear spin-lattice relaxation rate $1/T_1$ is plotted as a function of the inverse temperature on a semilog plot, for ${\text{BiCu}}_2{\text{PO}}_6$. The solid line is a fit to activated behavior as explained in the text. A few temperatures are shown on the top axis to provide clarity.

Figure 3

(Color online) ${}^{31}\text{P}$ NMR spectra for 2% Zn-doped ${\text{BiCu}}_2{\text{PO}}_6$ at various temperatures. The spectrum moves to smaller shifts and broadens as temperature is lowered.

Figure 4

(Color online) The ${}^{31}\text{P}$ NMR shift of ${\text{BiCu}}_2{\text{PO}}_6$ as a function of $T$ is not seen to be significantly changed by Zn or Ni doping while the bulk susceptibility (right axis in the inset) has a marked low-$T$ increase.

Figure 5

(Color online) The measured full width at half maximum (FWHM) of the ${}^{31}\text{P}$ NMR spectrum is plotted as a function of $T$ for undoped, Zn-doped (1% and 2%), and Ni-doped (1%) ${\text{BiCu}}_2{\text{PO}}_6$. As seen in the inset, the FWHM scales with the shift, except in the low-$T$ region. This has been used to extract the dopant-induced FWHM in the various samples, as explained in the text.

Figure 6

(Color online) The $T$ dependence of the impurity-only contribution to the width $\Delta {\nu }_{\text{impurity}}$ is plotted for different samples. The “intrinsic” width was subtracted from the measured FWHM as explained in the text.

Figure 7

(Color online) Schematic for the model in Eq. (2) of coupled ladders having three different couplings $J_{leg},J_{rung},J_{\perp }$. Focusing in the gapped regime $\left(J_{leg}=J_{rung}=10J_{\perp }\right)$, the system displays a VBS ground-state illustrated by rung singlets. Each nonmagnetic impurity (black dots) breaks such a rung singlet and releases a spin-1/2 degree of freedom that is exponentially localized around the vacancy with an AF pattern.

Figure 8

(Color online) Main panel: theoretical NMR spectra from QMC simulations performed for a simple model of weakly coupled $S=1/2$ ladders [Eq. (2)] with $x=2\mathrm{\%}$ of non-magnetic (Zn-like) impurities. Inset: full widths at half $\Delta {\nu }_{1/2}$ and quarter $\Delta {\nu }_{1/4}$ maximum versus $T$, rapidly growing below the spin gap $\Delta \approx 37\text{K}$ in qualitative agreement with experimental data.

Figure 9

(Color online) Temperature dependence of the uniform susceptibility computed using QMC for a $32\times 32$ sites system of weakly coupled ladders [model in Eq. (2)] with $J=10J_{\perp }=100\text{K}$. Blue dotted line shows the results of $\chi \left(T\right)$ obtained on the pure system while open circles show the knight shift computed for doped samples ($x=2\mathrm{\%}$ of $S=0$ Zn impurities). Experimental NMR Knight shifts are also shown for comparison (green squares).

Figure 10

(Color online) Magnetization profiles of an isolated spin-1/2 ladder with a single spin-$S$ impurity (depicted on the top) coupled with $J^{\prime }$ to the rest. QMC results obtained on $32\times 2$ samples at $T=J/100$ and $H=J/10$. Three cases are shown here: (i) $S=0$ (open circle, Zn case) and (ii) $S_{\text{imp}}=1$ with $J^{\prime }=J$ (blue diamond) lead to quantitatively similar features while the third case (iii) $S_{\text{imp}}=1$ and $J^{\prime }=-J$ (red squares) shows an enhanced effect.

Figure 11

(Color online) QMC results for the amplitudes ratio $S_0^{\text{Ni}}/S_0^{\text{Zn}}$ of the magnetization profiles (doped and opposite legs) versus the coupling $J^{\prime }$ between Ni and Cu.

Figure 12

(Color online) Ni effect on the spin ladder in case of a ferromagnetic (upper panel) or antiferromagnetic (lower panel) coupling with neighbor Cu. For a $J_4$ AF next-neighbor coupling, frustration is removed by Ni for the ferromagnetic case.

Figure 13

(Color online) The dominant hybridization paths between one ${}^{31}\text{P}$ (in blue) and its adjacent Cu (in red) through oxygen orbitals (in green) are represented. For symmetry reasons, there is no hybridization via paths $\alpha$ and $\beta$.

Figure 14

(Color online) Hyperfine couplings between P (blue) and Cu (red) are symbolized by the blue lines. In the presence of a Zn, the P labeled $i+3$ and $i+5$ probe only three Cu.

Figure 15

(Color online) The ${}^{31}\text{P}$ NMR line shapes for Zn-doped ${\text{BiCu}}_2{\text{PO}}_6$ samples at 300 K. The shoulders are identified as arising from ${}^{31}\text{P}$ nuclei near the Zn as explained in the text.

Figure 16

(Color online) The relative intensities of the two satellites shown in Fig. 11 for various Zn contents, measured at $T=300\text{K}$.