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Impurity effects in coupled-ladder BiCu2PO6 studied by NMR and quantum Monte Carlo simulations

We present a 31P NMR study of the coupled spin 1/2 ladder compound BiCu2PO6. In the pure material, intrinsic susceptibility and dynamics show a spin gap of about . Substitution of non magnetic Zn or magnetic Ni impurity at Cu site induces a staggered magnetization which results in a broadening of the 31P 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 (QMC) calculations assuming that Ni couples ferromagnetically to its adjacent Cu. The induced moment follows a 1/T temperature dependence due to the Curie-like 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 1D to 2D systems.


I. INTRODUCTION
Studying the effects of impurity substitutions in lowdimensional magnetic systems is a productive method to reveal the magnetic properties of the pristine, undoped materials as well as to probe the unusual and novel shortrange magnetic disturbance that a dopant/impurity might cause around it. In particular, nuclear magnetic resonance (NMR) experiments have been successful in unravelling the details of the impurity-induced shortrange and long-range effects via a measurements of the NMR lineshapes and their dependencies on temperature. 1 A direct comparison with theoretical predictions of the ground-state properties is then possible. The prominent systems which have been investigated are the S = ½ (gapless) Heisenberg Antiferromagnetic (HAF) chain Sr 2 CuO 3 2 , the S = ½ (spin-gapped) spin-Peierls chain CuGeO 3 3 , the S = 1 (spin-gapped) HAF Haldane chain Y 2 BaNiO 5 4 , the S = ½ (spin-gapped) 2-leg ladder SrCu 2 O 3 5,6 etc. From the theory side, impurity-induced effects in low-D antiferromagnets has been studied quite extensively as well 7-15 . Continuing in this endeavor, we have recently reported 16 the occurrence of a spin-gap behaviour in the 2-leg ladder compound BiCu 2 PO 6 . This was confirmed by recent neutron experiments 17 . The most studied spinladder compound (SrCu 2 O 3 ) shows a very small coupling between ladders accompanied by a very large coupling J = 2000 K between the Cu spins in the ladder, i.e. it consists of almost isolated ladders. BiCu 2 PO 6 on the contrary shows a smaller coupling in the ladder J = 100 K but comparatively larger coupling between the ladders 16,17 . This allows to probe the intermediate situation between isolated ladders and 2D-Heisenberg AF planes which should help understand how peculiar the ladder geometry is and what the properties of such coupled ladders are. In this spirit, we recently reported μSR and NMR results 18 showing that even a small amount of magnetic or non-magnetic impurity induces spin freezing at low temperatures. This behaviour was shown to scale remarkably well with that of other lowdimensional systems, independent of their geometry, such as the isolated ladder SrCu 2 O 3 , spin chains PbNi 2 V 2 O 8 or Spin Peierls chains CuGeO 3 .
Thus motivated, we have carried out an extensive 31 P NMR study on undoped BiCu 2 PO 6 as well as BiCu 2 PO 6 with non-magnetic Zn and magnetic Ni impurities doped at the Cu site to further explore such universal impurity induced features. The NMR shift and spin-lattice relaxation rate provide clear evidence for a spin-gap in the undoped compound. Our measurements clearly show an impurity-induced broadening of the spectra at low temperatures in the doped samples which is attributed to the staggered magnetisation around an impurity. This is further confirmed by Quantum Monte Carlo (QMC) simulations which demonstrate that this broadening is due to the Curie-like development of an induced spin near the impurity, whose extension saturates at low temperatures like in S=1 spin chains 4 . Similar behavior is observed for both magnetic Ni and non-magnetic Zn impurities, implying that both impurities induce a staggered magnetization with similar extension, despite the fact that Ni S=1 is shown from QMC to couple ferromagnetically to its neighboring Cu S=1/2 spins. These results show that impurities induce the same type of (spatially) alternating magnetic moments on the sites neighbouring the impurity leading to an effectively extended moment as in uncoupled ladders or spin chains. The effective moment extension is controlled by the pure magnetic correlation length which implies a universal mechanism for a 3D spin freezing at low temperature 18 .

II. TECHNICAL DETAILS
Polycrystalline samples of Bi(Cu 1-x M x ) 2 PO 6 (M = Zn, Ni) were prepared by standard solid state reaction techniques. While Zn shows complete solid solubility, Ni can be substituted up to x = 20%. The details pertaining to sample preparation and characterisation are given in Ref. 19 . For the purpose of NMR, we have mainly worked on samples with x < 0.03. 31 P NMR measurements were done in a temperature range of 1.2-300 K. 31 P nucleus carries a spin I=1/2 with 100% natural abundance and gyromagnetic factor γ/2π=17.254 MHz/Tesla. Echo pulse NMR was used, with typical π/2 pulse widths of 3 μs, delays between pulses of 80 μs, and a total repetition time of 100 msec at high temperature and up to a few seconds at low temperature when relaxation times get longer. The spectra were obtained by recombining successive Fourier Transform (FT) of the spin-echo signal obtained in a fixed field of 70 kOe by sweeping the frequency. The NMR total shift K was obtained from the relative shift of the peak of the 31 P NMR line with respect to that in an H 3 PO 4 solution used as a reference. The spin-lattice relaxation time T 1 was obtained using a standard saturation-recovery procedure. The recovery was found to be single exponential as expected for the I = ½ 31 P nucleus.

III. PURE BiCu 2 PO 6
The temperature dependence of the magnetic susceptibility of our undoped BiCu 2 PO 6 sample has been reported in Ref 16 . We note that the bulk susceptibility will have different intrinsic and extrinsic contributions which are generally difficult to disentangle. In particular, analysis of low-temperature data can be troublesome when a large Curie-like term is present. In contrast, the NMR shift is insensitive to the presence of small amounts of extrinsic impurities and it measures the intrinsic spin susceptibility. The 31 P nucleus is indeed coupled to the copper spins via a super-transferred hyperfine coupling A hf via the oxygen (see Appendix for details). The NMR shift K consists then of two parts : where chem K is the chemical shift, which could be measured here in BiZn 2 PO 6 . Indeed, BiZn 2 PO 6 shows no spin contribution because of the absence of Cu, but its structure and the P orbitals are isotructural to those of BiCu 2 PO 6 . This ensures that the T-independent shift of only 10 ppm measured in BiZn 2 PO 6 is directly chem K . This correction allows us to be confident about the "zero" of the shift (left) axis in Fig.1, where the spin shift 31 P K spin is compared to the measured susceptibility χ bulk in undoped BiCu 2 PO 6 as a function of temperature T.  It is seen from Fig. 1 that K scales with χ bulk . As was anticipated, there is a difference between the behaviour of K and χ bulk at low-temperature which is due to the presence of extrinsic contributions (as well as intrinsic defects/chain breaks) in χ bulk . On the other hand, K measures the intrinsic spin susceptibility which shows no low-temperature upturn. The prominent features of the temperature dependence of K are two-fold: (i) there is a broad maximum (at about 60 K) which is a hallmark of low-dimensional magnetic systems, and (ii) there is a sharp decrease below the maximum, with K becoming small as T approaches zero, due to a gap in the spin excitations. Fitting the shift data at low temperatures to which is valid for onedimensional spin-gapped systems, 20 yields for the gap Δ/k B . The reason for a non-zero residual value for K 0 (~ 200 ppm) is not clear at present but is not due to any chemical shift contribution as explained hereabove.
The nuclear spin-lattice relaxation rate 1/T 1 , which probes the low-energy spin dynamics in a system, has been known to provide important information in a variety of low-dimensional spin systems such as the spin chain Sr 2 CuO 3 21 , the two-leg ladder SrCu 2 O 3 22 , the Haldane chain Y 2 BaNiO 5 23 , etc. Our 31 P 1/T 1 data (see Fig. 2) show an activated exp(-Δ 1/Τ1 /k B T) behaviour as expected for a gapped system, and we find that The T-dependence of 1/T 1 in onedimensional spin-gapped systems 20 is indeed expected to follow ) limit, the leading behaviour of χ and 1/T 1 is going to be ) . This ratio is consistent with our findings within error bars. The deviation of 1/T 1 from the exponential decrease at low-T might be due to intrinsic defects.
So, both the static spin susceptibility and the dynamic relaxation rate NMR measurements consistently demonstrate that the system is spin-gapped, as expected in such a ladder geometry. 6 31 P NMR spectra were measured for Ni and Zn substituted BiCu 2 PO 6 at various temperatures. Typical evolution of the spectra is reported in Fig. 3. When temperature is lowered, the spectrum shifts toward lower frequencies and strongly broadens. When a local defect is created at a Cu site, NMR enables one to measure the histogram of the local fields both close to and far from the defect through 31 P NMR hyperfine coupling to all the Cu sites. As demonstrated in various low-dimensional AF chains, ladders, or planes, the center of gravity of the spectra, i.e. their shift, is proportional to the spin susceptibility far from the defects 1 . In contrast, any induced magnetism close to the impurity is reflected, either in a broadening of the spectrum or in the appearance of satellites in the wings of the spectrum. The former is exactly the behavior observed in Fig.3. A. Susceptibility far from the defects The 31 P shift plotted in Fig. 4 is found almost identical for substituted and pure materials. This is direct evidence that the uniform spin susceptibility χ 0 far from the defects is unaffected either by Zn or Ni. So the ladder spin-gap estimated to be 35-40 K in the pure compound remains identical in the presence of impurities. A very slight increase of the low-temperature shift is observed for Zn:2% (see the inset of Fig.4). This could be due to the presence of some in-gap, impurity-induced density of states but, given the experimental accuracy, the effect is too small to allow firm conclusions.

IV. NMR STUDY OF THE IMPURITY EFFECTS
A comparison of the 31 P NMR shift with macroscopic susceptibility can be made, using the hyperfine coupling estimated in pure BiCu 2 PO 6 , as reported in the inset of Fig.4. At temperatures lower than that of the broad maximum, down to T=25 K, the macroscopic susceptibility and the 31 P NMR shift both follow a similar decrease when decreasing temperature, due to the opening of the spin -gap. B. Induced staggered magnetization close to the defects At temperatures lower than T=25K, the NMR spin susceptibility far from the defects saturates to almost zero while the macroscopic susceptibility diverges, typically as 1/(T+θ). In the same low-temperature regime, the NMR linewidths increase with a similar Tbehavior, as shown in Fig.5.
This low-T Curie-like increase in macroscopic susceptibility and the NMR broadenings signal the appearance of induced, staggered, paramagnetic moments near each impurity. Both effects scale with the impurity content as expected. Similar features were observed in various compounds with nonmagnetic or magnetic impurities, such as spin chains, 2 spin ladders, 5 and 2D planes (high-T c cuprates). 1,24 For a quantitative comparison, we first have to correct the NMR linewidths for other sources of broadening. The inset of Fig. 5 shows that the linewidth and the shift scale with each other above T = 30 K, when the impurity effect is not too large. It means that the NMR line has an intrinsic, small width proportional to the shift. It is likely due to the random orientation of the crystallites composing our samples, and which in turn gives rise to a distribution of the shift itself. We used this scaling to extrapolate this "intrinsic" contribution to the width down to T = 0 and subtracted it from the full linewidth. The resulting impurity-only contribution to the width impurity ν Δ is plotted on Fig.6. As seen, it is sizeable only at low-temperatures and scales well with the impurity content, as expected. Note that a Curie behavior is also observed for the width of the pure compound, which implies that there are some intrinsic defects in the material such as vacancies or other types of local disorders, as is usual in these oxides 2 . This allows us to estimate the typical amount of such native defects to be about x=0.1 to 0.3%, a value which agrees well with macroscopic measurements reported on the same samples. 19  The T-dependence of the impurity-only contribution to the width Δν impurity is plotted for different samples. The "intrinsic" width was subtracted from the measured FWHM as explained in the text.

V. QUANTUM MONTE CARLO VERSUS NMR
QMC is a very precise tool to investigate localized moments physics in spin gapped materials like spin ladders 25 , Haldane chains 26,4 , or coupled ladders systems 27 . Here, we employ Stochastic Series Expansions (SSE) method 28,29 to get a better understanding of how staggered moments develop in a coupled ladders model perturbed by a finite concentration of spinless (Zn) or S=1 (Ni) impurities and how this modifies NMR spectra.

A. The model
The starting point is a realistic model of AF coupled spin-1/2 ladders that we previously investigated in Ref. 18 : . . . Muffin-tin orbital calculations suggest the existence of an additional frustrating coupling along the ladders 16 . We did not take this possible frustration into account in our model Eq.
(2) as we fo c u s o n c a p tur in g semiquantitative features of the low-T physics i.e. a weakly coupled ladders system having a sizable spin gap. The system of coupled ladders with static spinless impurities is depicted in Fig.7 are computed for each randomly doped sample.
Once data from a sufficiently large enough number of independent samples are obtained, an histogram of the Knight Shifts P(K i ) can be drawn (see Fig. 8) with no additional free parameter, using Eq.(5) which relates the 31 P NMR shift and the local magnetizations. While quite simple, the model Eq.
(2) captures qualitative and even semi-quantitative features of experimental spectra where one can observe (Fig. 9) shift and broadening temperature dependences similar to the experimental ones. As a first theoretical check, we have compared the Tdependence of the average shift position K(T) (which represents the magnetic response far from the defects) with the spin susceptibility of the pure system χ(T) computed independently with QMC on a pure system. These results are shown in Fig. 9 where, as expected, a good agreement is observed between theoretical K and χ. Both K and χ are also consistent in magnitude and behavior with the experimental NMR shifts (green squares) but show deviation from the experiment at intermediate temperature around the maximum and at higher temperatures. Such a discrepancy is attributed to frustration effects, not included in our theoretical model, which are known to reduce the maximum of χ 32 .

C. Impurity effects from QMC simulations
The broadening of the local fields distribution is a direct signature of the impurity-induced paramagnetic effects in a magnetic field when the system is cooled down below the spin gap ∆. This broadening is observed in the QMC spectra at low T as seen in the inset of Fig. 8 where the full widths at half ∆ν 1/2 and quarter ∆ν 1/4 maximum rapidly grow at low T, in qualitative agreement with experiments ( fig.6). However, our simple 2D model does not allow for a direct quantitative comparison between QMC and NMR since frustration and, more importantly, finite-T g 3D ordering are not included in the model. Beyond these discrepancies, we can discuss the general broadening mechanisms at play in the real 3D material.
For T ≤ ∆, exponentially localized induced moments start to build up in the vicinity of each dopant. For a 3D system the average moment profile displays the 3D localized form are the correlation lengths for AF order in the three spatial directions. They are in general all different due to the anisotropy of interactions. The effect of the impurity is to induce an effective moment of typical size S 0 with a 3D spatial extension z y x V ξ ξ ξ ξ ∝ around the impurity. In the dilute limit ( x V / 1 << ξ ), we expect the width of the local fields distribution to scale with the impurity concentration x, the typical size of the induced moment S 0 and its 3D spatial extension V ξ : . (4) This relation can be deduced intuitively in the following way. The broadening is due to unresolved satellites corresponding to the induced moments on spins close to the impurity. Their intensity scales with x and their number scales with V ξ . These are not resolved here because of the dimensionnality and hyperfine factor of 31 P which averages over 4 sites. Two main sources of broadening are therefore showing up upon cooling of the system: a rapid growth of (i) the extension and (ii) the size of induced moments. (i) Regarding the spatial extension ) (T V ξ , while a rapid increase is expected below T~∆, a transient saturation regime appears before 3D effects at lower T come into play, formally leading to a diverging V ξ (T) when T→T g .
Note however that strongly anisotropic effects are also expected, leading to a more complicated T -dependence of ) (T V ξ . (ii) Concerning the typical size 0 S of the induced moments, below the spin gap there is a free impurity spins regime 8 where the external field induces a Brillouin response which leads to a Curie-like behavior T H S / 0 ∝ , in the range ∆>T>>H, and saturates at lower T. Note however that frustration combined with random dilution may also lead to a reduction of the effective moment at lower temperature 33 . Taken all together, these various effects imply that the exact temperature (and field) broadening of the NMR lines is a quite subtle issue that will be investigated further in a future work.

D. Comparison between Zn and Ni
We now turn to the difference between non-magnetic Zn and magnetic S=1 Ni-doped samples. As shown in Fig.6, there is roughly a factor 2 between experimental NMR linewidths ( 9 ). However, the spatial extension of induced moments does not depend on the magnetic nature of the impurities, as shown in Fig.10. Indeed, such an extension is basically controlled at low enough temperature by the inverse bare spin gap 1/∆. Therefore this difference has to come from the moment size itself 0 S . Such a conclusion is also reached by the analysis of Curie tails performed in Ref. 19  In order to understand the physical origin of such an effect, we performed QMC simulations on a simple model of isolated ladders and compared magnetization profiles for 0 = imp S or 1 = imp S coupled to the bulk spins-1/2 with a coupling J' that may be different from J (depicted on top of Fig. 10). Results of such a computation are shown in Fig. 10 for three representative cases: for an antiferromagnetic AF J'>0 and a ferromagnetic F J'<0 coupling with Cu. As stated above, the induced moment extensions are roughly identical: Zn Ni ξ ξ ≈ . However, the prefactor 0 S of the exponential decay does depend on the nature of the coupling J' between Cu (S = 1/2) and Ni (S = 1). is ferromagneticaly coupled to its neighboring spins 1/2, the ground-state is in the subsector S z =3/2 which leads to an enhancement of the magnetization profile, without changing the spatial extension ξ. To go further, we have performed systematic QMC computations for the magnetization profiles on isolated ladders for 1 = imp S and various J' and compared the results to the 0 = imp S case. After performing an exponential fit, we get the amplitudes Ni S 0 plotted in Fig. 11 versus J'. The experimentally observed factor 2 is obtained here for a ferromagnetic coupling J'of the order of J at least. Note that simulations are performed on isolated ladders. In the real compound, we expect dimensionnality and frustration effects to enhance correlations and therefore 0 S , leading to a smaller absolute value of J' to account for the experimental findings. While it is hard to check these findings from simple quantum chemistry arguments, we stress that frustration combined with a ferromagnetic coupling between Ni and Cu will naturally enhance the effective moment amplitude around a Ni impurity. Indeed, as schematized in fig.12, frustration which exists in pure BCPO because of a second neighbor AF coupling J 4 between Cu would be removed here locally near Ni. Such an effect will naturally enhance local moment amplitudes. Therefore, while it is hard to get a precise estimate of how Ni couples to the neighboring Cu in Ni-doped BiCu 2 PO 6 , the QMC simulations and NMR experiment suggest a substantial ferromagnetic coupling J' of the order of J.

VI. DISCUSSION
We have shown that in the coupled spin ladder BiCu 2 PO 6 , non magnetic and magnetic impurities both induce staggered moments which develop below the spin gap. Their extension is similar, only their amplitude differs, because Ni couples ferromagnetically to its neighboring Cu, while Zn does not couple at all. These results can be compared to similar studies in other low dimensional quantum magnets.
In isolated spin ladders SrCu 2 O 3¸ Cu NMR studies reveal similar qualitative behavior: both Zn and Ni induce Curie-like paramagnetic NMR broadenings which signal staggered moments 5,6 . But opposite to our case, Ni is found to induce a broadening identical to Zn 6 . In these isolated ladders, Ni is indeed probably coupled to its adjacent Cu with an AF coupling similar to that of the pure ladder, leading to same effect than Zn. This contrasts with BiCu 2 PO 6 where the likely ferromagnetic effective Ni-Cu couling, in concomitance to large inter ladders couplings and possibly frustration effects, produces an induced moment larger for Ni than for Zn. While qualitative results are similar in our study and Ref. 5 , interpretations are very different. In Ref. 5 , it has been argued that the moment extension ξ depends on impurity content as ξ~1/x. This would contradict our own interpretation and QMC results where ξ is found to be independent of impurity content, and governed solely by the intrinsic correlation of the pure ladder. But this ξ~1/x conclusion comes mostly from the data taken at a very small Zn content x=0.1% where broadening is not scaling with impurity content anymore. However, at such small content, the pure linewidth and the impurity effect are so small that incertitude is too large to determine the impurity-induced broadening safely. All other data of Ref. 5 , and all data reported in an independent paper 6 on same compounds are consistent with our own interpretation of ξ independent of impurity content and intrinsic to the ladder's physics.
In spin S=1 Ni Haldane chains, the very same trends are observed: both non magnetic Zn and magnetic S=1/2 Cu at Ni site induce similar exponentially decaying staggered moments, with same extension ξ and slightly different amplitudes 4 , ξ being hereagain identical to the pure correlation length.
Finally, underdoped high Tc cuprates such as YBa 2 Cu 3 O 6+y (0.4<y<0.9) can be considered as a realization of a strongly antiferromagnetically correlated two dimensional Cu system, even though they are also metallic. Here again, both non magnetic Zn or magnetic Ni substituted at Cu site of CuO 2 two-dimensional S=1/2 layers induce similar staggered moments with the same temperature dependencies but different amplitudes 1,24 .
Our results suggest a possible universal picture which could apply to quantum antiferromagnetically coupled low dimensional systems, the BiCu 2 PO 6 bridging the 1D physics of spin chains and isolated ladders to the 2D physics of cuprates. Any non magnetic as well as magnetic impurity induces a staggered moment at temperatures smaller than the relevant energy scale (magnetic coupling and/or spin gap). This moment follows a Curie-like dependence which may be affected if the extension ξ also depends on temperature. Only the amplitude differs in the case of a magnetic impurity, depending on the size and sign of the coupling between this impurity and neighboring spins. The extension ξ is characteristic of the intrinsic system and does not depend on impurity content at small impurity content. At low enough temperature, because of the other residual 3D couplings present in the compound, these induced extended moments may interact enough to freeze in an antiferromagnet or a cluster-type antiferromagnet, in a universal fashion as well. This generic scenario needs to be further confirmed in other low dimensional systems with different geometries and couplings.
Precise understanding of how this generic 3D ordering takes place remains an open issue related to the important question of criticality close to a transition, and needs more theoretical and experimental effort.

APPENDIX : HYPERFINE COUPLINGS IN BiCu 2 PO 6
In this appendix, we demonstrate from NMR experiments that each P nucleus is coupled through hyperfine coupling to two Cu atoms of the adjacent ladder as shown in fig.14 . The spin density largely resides on the Cu orbitals and an effect (in terms of an NMR shift) is felt at the 31 P site due to a hybridisation of the P 4s orbitals with the Cu d orbitals via the oxygen. We want to determine which hybridization actually contributes to this coupling. The various possible hybridization paths between 31 P and adjacent Cu are plotted on Fig.13. The band structure calculations reveal that in the global (crystallographic) co-ordinate system the Cu yz orbitals lie near the Fermi level, so we focus on this orbital only and its effective hybridization with P−s orbitals 35 . Because of symmetry reasons, this hybridization along path labelled α and β is zero. Therefore, each P is coupled to four Cu all from the same ladder through the path labelled a, as schematically displayed in Fig. 13.  This results for each P nucleus at site i in a shift: is the hyperfine coupling between one 31 P and one Cu via path a. This can be checked experimentally from the analysis of Zn substitution effect on the 31 P NMR spectrum. When the temperature is much larger than the gap opening, no induced moment is expected from Zn substitution at the Cu site. For one Zn defect, the only effect is the removal of one Cu and of its contribution to the near neighbour (nn) 31 P shift coupled to this Zn (Fig.14). If each 31 P is coupled to N Cu, the P nn to Zn should lead to a distinct less shifted satellite line, with a shift The 31 P NMR spectrum at T=300K plotted in Fig.15 indeed displays these two reduced and less shifted satellites in the presence of Zn. The measured satellite shifts nn K and * nn K are found reduced compared to K main by a ratio 0.7 and 0.4 respectively which points to N between 3 and 4. But Zn could slightly distort its environment and change the local hyperfine couplings, which would in turn affect these ratios. On the contrary, the intensities of these satellites plotted on fig.16 depend only on the Zn concentration x which makes it a more reliable quantity. A good agreement with N=4 is found for both satellites, with no adjustable parameter. This unambiguously confirms that the picture depicted in Fig.14 is correct, where each P is coupled to the 4 Cu of a single ladder. The hybridization ε between Cu d and P 3s orbitals at play in the hyperfine coupling path a can be roughly estimated using :