Quantum Monte Carlo simulations for stacked spin-ladder systems containing low concentrations of nonmagnetic impurities: Application to the low-temperature broadening of NMR spectra in SrCu${}_2$O${}_3$

We present a quantum Monte Carlo study for Heisenberg spin-$\frac{1}{2}$ two-leg ladder systems doped with nonmagnetic impurities. The simulations are applied to the doped spin-ladder compound Sr(Cu${}_{1-x}$Zn${}_x$)${}_2$O${}_3$, where a large broadening of the ${}^{65}$Cu NMR lines has been observed in experiment at low temperatures but far above the Néel temperature. We find that interladder couplings with a sizable coupling in the stacking direction are required to describe the line broadening, which cannot be explained by considering a single ladder only. Around a single impurity, spin correlations cause an exponentially decaying antiferromagnetic local magnetization in a magnetic field. We develop an effective model for the local magnetization of systems with many randomly distributed impurities, with few parameters which can be extracted out of quantum Monte Carlo calculations with a single impurity. The broadening arises from a drag effect, where the magnetization around an impurity works as an effective field for spins on the neighboring ladders, causing a nonexponentially decaying magnetization cloud around the impurity. Our results show that even for impurity concentrations as small as $x=0.001$ and $x=0.0025$, the broadening effect is large, in good quantitative agreement with experiment. We also develop a simple model for the effective interaction of two impurity spins.

Figure 1

Experimental NMR spectra from Fujiwara et al. (Ref. [3]). The right panel shows the NMR linewidth broadening in doped SrCu${}_2$O${}_3$, while for the undoped sample (left panel) this effect is almost absent.

Figure 2

Model structure for SrCu${}_2$O${}_3$. Left panel: Trellis lattice in the Cu-O plane. The effective ladders (solid lines) are decoupled by frustration, symbolized by the dashed lines. We denote by $S_0$ (cross) a spin located on the same rung as a nonmagnetic impurity (open circle). Right panel: Stacked ladders coupled via $J_3$.

Figure 3

Rung picture of a spin ladder without (left) and with (right) an impurity. Spins form singlets with a spin gap which prevents the system from responding to a small external field. An impurity breaks one singlet and leaves a free spin-$1/2$ which responds to small external fields and couples to surrounding spins.

Figure 4

Local spin expectation value (absolute value) of a single ladder for $J_R/J_L=0.5$, $H=0.01J_L$ with a single impurity, at two different temperatures. Upper curve (red/blue) $T=0.02J_L\simeq 40$ K, lower curve (black/green) $T=0.05J_L\simeq 100$ K. The different colors refer to magnetic moments located on the undoped ladder leg (red and black circles) and on the doped ladder leg (blue and green triangles), respectively. For the two temperatures considered, the correlation length remains almost constant (lower inset). The upper inset shows the spin expectation values at $T/J_L=0.02$ for different values $J_R/J_L$. The correlation length ${\xi }_x(J_R/J_L)$ becomes $5.9$, $7.45$, and $9.75$ for $J_R/J_L=$ $0.6$, $0.5$, and $0.4$, respectively.

Figure 5

(a) Total magnetization versus $T/J_L$ for a system in a magnetic field, without (blue up-triangles) and with (black circles) an impurity. System parameters: 200 $\times$ 2 ladder, $J_R$/$J_L=$ 0.5, $H=0.005J_L$ ($\simeq 14.3$ T). To observe the effect caused by the impurity only, the difference between the undoped and doped systems is also shown (green down-triangles). The red dashed line is the analytical solution for a free spin in a magnetic field. Blue and black lines are guides to the eye. In the insets, the staggered distributions of local magnetic moments around the impurity at (b) $T=0.005J_L\simeq 9.6$ K and (c) $T=0.09J_L\simeq 173$ K are plotted. (For each inset: ladder leg with impurity on the right, leg opposite impurity on the left.)

Figure 6

Spin expectation values of the spins opposite of the impurity ($i=0$) and at a distance of 10 lattice sites ($i=10$) and 20 lattice sites ($i=20$) versus $T/J_L$, all on the ladder leg opposite the impurity. The dashed lines correspond to a free spin in a magnetic field times a proportionality factor. System parameters: 200 $\times$ 2 ladder, $J_R$/$J_L=$ 0.5, $H$/$J_L=$ 0.005. The inset shows the factor $A\left({\xi }_x\right)$, measured at $T=0.02J_L$.

Figure 7

Total magnetization versus $H/J_L$ for a system with a single impurity (blue up-triangles). System parameters: 300 $\times$ 2 ladder, $J_R$/$J_L=$ 0.5, $T_{\mathrm{ref}}$/$J_L=$ 0.02083. The red dashed line corresponds to the solution of a free spin in a magnetic field. The inset compares Eq. (4) with the QMC results for $\langle S_{i,0}^z\rangle$, with $A\left(\xi \right)$ measured at $H_{\mathrm{ref}}=0.01J_L$ and the same $T_{\mathrm{ref}}$.

Figure 8

Magnetic moment profiles with two impurities on the same ladder leg. Full lines: description according to Eq. (6), symbols: QMC results. Upper curve: impurities are located at positions 123 and 137 on the same sublattice. Lower curve: impurities are located at positions 123 and 136 on different sublattices. QMC simulation for $J_R/J_L=0.5$, $H_{\mathrm{ref}}/J_L=0.01$, $T_{\mathrm{ref}}/J_L=0.02083$.

Figure 9

${\langle S_0^z\left(d\right)\rangle }_{2\mathrm{imp}}$ versus distance of two impurities on the same ladder leg. QMC results (black error bars) are compared to an effective two-site Heisenberg model (red triangles). The blue dotted line corresponds to a simple exponential superposition of two magnetic moment profiles calculated from a single impurity. ($T/J_L=0.02083,H/J_L=0.01$, system size $200\times 2$.)

Figure 10

Absolute values of the local spin expectation values induced by a single impurity ($\triangleq 0.08\dot{3}\%$) for different $J_3$/$J_L$ with $J_R$/$J_L=$ 0.5. The system size is 100 $\times$ 2 $\times$ 6 spins with periodic boundary conditions in leg and stacking direction. $T=0.02083J_L\simeq 40$ K and $H=0.01J_L\simeq 30$ T. The impurity resides on the right leg ($j=1$) of ladder 1 ($k=0$). For the magnitude of spin expectation values see the inset, where the legs with $j=0$ of the same three stacked ladders are plotted at $J_3/J_L=0.019$ on a logarithmic scale, demonstrating the nonexponential behavior found on neighboring ladders.

Figure 11

$|\langle S_{0,0,k}\rangle |$ for a stacked ladder system ($k=0,1,2,...,7$). $J_R/J_L=0.5$, $H/J_L=0.01$, $T/J_L=0.02083$ and (from bottom to top) $J_3/J_L=0.01,0.015,0.02,0.025,0.03$. The inset compares the QMC results for $J_3/J_L=0.02$ to a hyperbolic cosine, demonstrating the nonexponential behavior. Lines are guides to the eye.

Figure 12

Local spin expectation values (absolute values) for 6 stacked ladders with 3 impurities. Spin sites 1–200 correspond to ladder 1, 201–400 to ladder 2, and so on. The red line (open circles) is the result of our model function; the underlying blue solid line is the corresponding QMC simulation. Upper panel: impurities located on 3 different ladders [$(27,0,0)$, $(50,0,1)$, $(46,0,4)$]. Lower panel: 2 impurities located on the same ladder [$(44,1,4)$, $(92,1,4)$] and one on a different ladder [$(59,1,5)$]

Figure 13

Relation between local magnetic moment profiles and the corresponding NMR spectrum. Left panel: Spin expectation values (left and right leg) of a ladder containing two impurities on the left leg. Right panel: The corresponding NMR spectrum reflects the histogram of spin expectation values. The experimentally observed broadening corresponds to small values of $\langle S_{i,j,k}^z\rangle$ up to about 0.01.

Figure 14

NMR spectra with three impurities at fixed positions. Left panel: Influence of the stochastic noise on the width. Impurities at positions $(7,1,2)$, $(51,2,6)$, $(73,1,5)$, with QMC simulations of 10k (red dashed line) and 500k (black solid line) sweeps. The inset shows the dependence of the FWHM on the number of steps. Right panel: Two NMR spectra of two systems only differing in impurity configuration (250k steps). Black solid line: same as left panel; blue dashed line: positions $(35,1,1)$, $(52,1,4)$, $(86,2,5)$. ($J_R/J_L=0.5$, $J_3/J_L=0.017$, $T/J_L=0.02083$, $H/J_L=0.003828$.)

Figure 15

Simulated NMR spectra for $x=0.25\%$ impurities at $T=40$ K, $J_L=1920$ K, and $J_R/J_L=0.5$ with stacked ladder couplings $J_3/J_L=0.01$ (red dot-dot-dashed line) and $J_3/J_L=0.03$ (solid red line), compared to experiment (Fig. 1) (black dashed line). Sizeable broadening occurs only for the larger interladder coupling. At high temperature (340 K), all spectra are narrow. For reference, we show an undoped system (green dotted line, Gaussian line shape) and the doped system (blue dot-dashed line) at $J_3/J_L=0.03$.

Figure 16

FWHM vs temperature for a stacked ladder system with $J_L=1920$ K, $J_R/J_L=0.5$, and varying interladder coupling $J_3$/$J_L$: 0.0 (black diamonds), 0.01 (red down-triangles), 0.02 (green up-triangles), and 0.03 (blue squares). The black crosses show results with a different rung coupling $J_R/J_L=0.4$ at $J_3/J_L=0.02$. Experimental values for the ${}^{65}$Cu central peak from Fig. 1 are given by red open circles. The inset shows an expanded temperature range.

Figure 17

Simulated NMR spectra at $x=0.0025$, with $J_L=1920$ K, $J_R/J_L=0.5$, and $J_3/J_L=0.03$ and different temperatures compared to experiment (Ref. [3]) (${}^{65}$Cu left peak, ${\nu }_{\mathrm{RF}}=83.55$ MHz, filled circles).

Figure 18

Simulated NMR spectra at $x=0.001$, with $J_L=1920$ K $J_R/J_L=0.5$, and $J_3/J_L=0.03$ and different temperatures compared to experiment (Ref. [4]) (${}^{65}$Cu left peak, ${\nu }_{\mathrm{RF}}=125.1$ MHz, filled circles). The asymmetric experimental profiles below 40 K are caused by an overlap of a ${}^{63}$Cu transition with its main peak around 10.05 T. The experiment appears to have a larger natural linewidth than assumed in our simulations.

Figure 19

Simulated NMR spectra (dashed lines) at $x=0.01$, with $J_L=1920$ K, $J_R/J_L=0.5$, and $J_3/J_L=0.03$ and different temperatures compared with experiment (Ref. [4]) (${}^{65}$Cu left peak, filled circles). The inset shows the same system with an impurity concentration of $x=0.02$.