Tuning the critical magnetic field of the triplon Bose-Einstein condensation in Ba$_{3-x}$Sr$_x$Cr$_2$O$_8$

The structure and magnetic interactions of the triplon Bose-Einstein condensation candidates Ba$_3$Cr$_2$O$_8$ and Sr$_3$Cr$_2$O$_8$ have been studied thoroughly in the literature, but little is known about a possible triplon condensation in the corresponding solid solution Ba$_{3-x}$Sr$_x$Cr$_2$O$_8$. We have prepared various members of this solid solution and systematically examined their magnetic properties in high magnetic fields up to 60 T and at low temperatures down to 340 mK, by means of pulsed field and cantilever magnetometry. From these experiments for $x\in\{3,2.9,2.8,2.7,2.6,2.5\}$, we find that the critical fields of Ba$_{3-x}$Sr$_x$Cr$_2$O$_8$ decrease monotonically as a function of the Sr content $x$. This change is in good agreement with the earlier reported variation of the magnetic interactions in these compounds.


I. INTRODUCTION
Over the past two decades, spin dimer systems have been in the focus of intense research due to their exotic magnetic properties. Several spin dimer materials have been shown to exhibit a spontaneous increase of the magnetization upon cooling below a certain temperature dimers that feature a dominant intradimer interaction constant J 0 and weaker interdimer interactions J . The crystal structure at room temperature can be described using the highly symmetric space group R3m 6,12 which leads to magnetic frustration of the spin system. Both materials undergo a Jahn-Teller induced structural phase transition upon cooling that lifts this magnetic frustration and strongly modifies the magnetic interactions in the system, thereby strengthening J 0 . 12 As no calibration of the pick-up signals has been performed, χ raw is not equal, but only proportional to the true sample susceptibility χ. However, for reasons of simplicity and as no absolute values of M and χ are used in this work, we refer to χ raw as χ in the following. Measurements of the empty magnetometer did only yield a smooth, featureless background χ BG . Thus, no background correction was applied to the resulting data for any of the examined samples. For fields above 40 T, the experimental noise was too large to perform a reliable numerical differentiation. Thus, only data for magnetic fields below 40 T have been used for our analysis and are shown in this work.

C. Cantilever magnetometry
The M (H) data at temperatures below T = 1 K were obtained from cantilever magnetometry experiments for samples with x = {2.9, 2.8} at the High Field Magnet Laboratory (HFML) in Nijmegen. The used method is based on measuring the change of the capacitance between a reference plate and a BeCu cantilever with the sample attached as a function of the magnetic field. This change is due to a slight bending of the cantilever by the torque exerted on the sample in an external magnetic field. The capacitance was examined using a Andeen-Hagerling AH 2700A capacitance bridge and a Stanford Research SR830 lock-in amplifier and the field was applied perpendicular to the cantilever. As the samples were polycrystalline and thus very isotropic, no torque can be measured in the center of the field. Accordingly, the samples were placed outside the field center where the field exhibits a certain gradient. To improve the sensitivity at temperatures below 2 K, the samples were placed between 1 cm and 3 cm above the field center, depending on the desired maximum field.
Due to the geometry of the experiment, the magnetic moment M of the sample can be During several test measurements, the capacitance change was found to be antisymmetric with respect to the field center when placing the sample below or above. Thus, we are certain that changes of the capacitance signal are only due to variation of the sample magnetization.
Similar to the case of the pulsed-field data, no background subtraction was conducted, as the BeCu cantilever itself only gives a negligible signal.

D. Analysis
As in the case of pure Sr 3 Cr 2 O 8 7 , no hysteresis was found upon reversing the direction of the variation of H for the triplon phase transition. Thus, our data analysis is based on the assumption that the triplon BEC is a second order phase transition, as expected.
This implies a discontinuity in the second derivatives of the Gibbs free enthalpy 17 . As the magnetization is the first derivative of the Gibbs enthalpy with respect to the external field, The position of the peak in ρ(H) is usually taken as the critical field H c (see below). 18,19 The derivatives were numerically obtained as difference quotient with subsequent smoothing through a symmetric running average. The smoothing window was kept smaller than 40% of the full width at half maximum of the observed peak in ρ(H), so that no significant additional broadening was introduced.
Determining the position of the peak in ρ(H) from the maximum value of the second numerical derivative of M (H) did not yield reliable results due to the significant noise level.
The thus determined values of the critical field depended strongly on the chosen smoothing window. We have therefore decided to fit the peak in ρ(H) using a analytical function and determine H c from the maximum of this function. A common choice for fitting the peak in ρ(H) is a Gaussian function with a symmetric shape. 20 However, it became clear that our ρ(H) data are always slightly asymmetric with a tail towards lower fields, even for pure Sr 3 Cr 3 O 8 (see, e.g., Fig. 1). This kind of asymmetry is not exclusive to Ba 3−x Sr x Cr 2 O 8 , but can also be found in the compound NiCl 2 ·4SC(ND 2 ) 2 . 20 . In our system, it becomes much more pronounced at high temperatures and especially for intermediate values of the Sr content x. This temperature dependent asymmetry can be accomodated for by convoluting the Gaussian function by a temperature dependent exponential: where erfc(x) is the complementary error function. The resulting fit to our data (with free parameters H 0 (x, T ), σ(x, T ) and α(x)) is excellent (see Fig. 1) and allows us to reliably determine the maximum of ρ(H) using numerical methods.
The critical fields H c were then taken as this maximum. The uncertainty of H c was defined by allowing the sum of the squared residuals, In the case of Ba 3−x Sr x Cr 2 O 8 , we attribute this broadening to the increasing disorder that has been reported for this system. 11 As described above, a change of the critical field can be induced by a corresponding variation of the magnetic interactions in the system. Such a variation of the intradimer interaction constant J 0 has been reported for Ba 3−x Sr x Cr 2 O 8 due the partial replacement of Ba by Sr. 10 In Fig. 2, we compare this change with the variation of H c as a function of x.
As the trends for J 0 (x) and H c (x) coincide well for all examined values of x, we conclude that the observed decrease of the critical fields at low temperatures can probably mainly be to free spins is too large in our samples to draw any decisive conclusion regarding the low field susceptibility of our samples, however. In any case, the magnetic phase boundary for a Bose-Glass of triplons should be tangential to the field axis for T → 0, 22,25 rather than perpendicular to it for a three-dimensional triplon BEC. Our data do not indicate such a behavior although measurements at even lower temperatures would be necessary for absolute confidence.
For temperatures above T ≈ 3.5 K, our experimental ρ(H) data show peaks that do not fit to the expected dome-like shape of the phase boundary for the triplon-BEC transition. These features, marked by a straight dashed line in Fig. 3, are reminiscent of a phase boundary or crossover other than the appearance of a long range XY -order of the spin system. Such a behavior of M (H) has also been found in single crystals of pure Sr 3 Cr 2 O 8 . 26 It has been suggested 27 that the closing of the spin gap can lead to similar features, e.g. in the T (H) traces when measuring the magnetocaloric effect of a gapped spin system.
We would like to point out that determining H c from the maximum of the observed ρ(H) of our samples is the common choice, but it may be not the only one. As described above,