Chemical ordering suppresses large-scale electronic phase separation in doped manganites

For strongly correlated oxides, it has been a long-standing issue regarding the role of the chemical ordering of the dopants on the physical properties. Here, using unit cell by unit cell superlattice growth technique, we determine the role of chemical ordering of the Pr dopant in a colossal magnetoresistant (La1−yPry)1−xCaxMnO3 (LPCMO) system, which has been well known for its large length-scale electronic phase separation phenomena. Our experimental results show that the chemical ordering of Pr leads to marked reduction of the length scale of electronic phase separations. Moreover, compared with the conventional Pr-disordered LPCMO system, the Pr-ordered LPCMO system has a metal–insulator transition that is ∼100 K higher because the ferromagnetic metallic phase is more dominant at all temperatures below the Curie temperature.


Supplem
(black), warming shows t under z than tha  The x-ray diffraction data was analyzed with Rietveld method based on the program FULLPROF 1 . We find that the diffraction spots of 2:1 superlattices are not exactly at 1/3 and 2/3, but aligned well with the integer index positions for both O-LPCMO and R-LPCMO (Supplementary Figure 1a and b). Here we derived the x-ray diffraction angle of the O-LPCMO film using a simple three layer model.

MFM images 2 T) Co/Pt
Our finding is that when the number of layers in the superlattice is finite, the diffraction spot shifts from the (001) and (002) positions. The features of the shifts (direction, magnitude) turn out to be useful in determining the detail of the superlattice.

Model
We use a three layer model consisting only La and Pr atoms. All the other atoms are omitted for simplicity. The atomic positions are: 0,0,0 , 0,0, , 0,0, , where is the lattice constant of the super cell. Note that we use the large 2La+1Pr unit cell for the lattice indexing.
The diffraction intensity from the superlattice follows: where , are the indices for atoms and unit cells respectively. Since , , one can rewrite Supplementary Equation 1 as where . The first term ∑ ⋅ is often called structure factor which determines the diffraction intensity; the second term ∑ ⋅ normally determines the diffraction angle. Below we discuss the two terms separately and show that due to the finite size effect, the diffraction maximum may not be at the (001) and (002) positions.

Finite size effect
First, we look at the factor ∑ ⋅ . One can derive the sum analytically, the result is (3) where is the total number of unit cells. This function corresponds to main maxima at 2 , or . There are also values for which are zero (minima). These values corresponds to 2 or ( ). There are also satellite maxima between those minima.
When is infinite, only the main maxima is important. In this case, the diffraction peak width is very sharp. When N is finite, the satellite maxima are also important and the diffraction peaks can be broad.
Next, we show that the dependence of the structure factor on may shift the main maxima.
The structure factor ∑ ⋅ for the LPCMO system can be written as Let's discuss diffraction at 001 . In this case, the factor is a maximum due to Supplementary Equation 3. On the other hand, is not at maximum. In other words, does not have a maximum at 001 .

Shift of diffraction peaks
We can further discuss which way the maximum shifts at 001 . Since the value of is symmetric with respect to around (001), one just have to find how change with respect to .
We calculate sin sin .
At 003 , 2 ; this means that 0. Therefore, the maximum intensity occurs at 003 .

Compare with experiments
We read the peak positions from our experiments as shown below (using STO and index LPCMO indices).
Supplementary The (001) sequence shift toward larger direction; the (002) sequence shifts toward smaller k direction; all the (003) sequence are well aligned. This means that the experiments and the model are consistent.

Quantitative analysis
We can estimate the intensity analytically using the model.

a) Direction of the shift
Note that when we analyze the slope of , the factor is important for the sign. Our experiments are consistent with the fact that the ordering is LaLaPr according to the shift of the diffraction peaks. Important implication is that if the ordering is LaPrPr, we will be able to see the peaks shifting toward the other direction.

b) Magnitude of shift
Another feature of the shift is the magnitude. One can see that for both and , if we replace with 3 , the values remain the same. So the model predicts that the peak shifts are the same for the same sequence, as observed by the experiments.

Conclusion
If we use the simple three-layer model, all the features of the observed XRD peaks can be explained. The only problem is that the predicted number of layers is too small (11)(12). This could be due to the shift atomic positions in the PCMO with respect to those in the LCMO.

Supplementary Note 2: The reciprocal space mapping of O-LPCMO and R-LPCMO
The reciprocal space mapping (RSM) of the two sample are shown in Supplementary Figure 3. In this paper, the pseudocubic in-plane strain (assuming ) and out-of-plane strain are defined by 2 : The Poisson's ratio ν is defined by:

Supplementary Note 3: Fitting the X-ray refraction of O-LPCMO and R-LPCMO
We observed thickness fringes near the main sample peaks for both random alloyed LPCMO and superlattice samples in our x-ray diffraction data. In order to get precise thickness value, we also measured the x-ray refraction (XRR), as shown in Supplementary Figure 4. By fitting peaks from the XRR data 4-6 , the total thickness of R-LPCMO and O-LPCMO are about 61.6 nm and 61.2 nm, respectively.
From our R-LPCMO fitting data, there are three parts in the R-LPCMO film. The top part is the surface layers exposed to air (~4-5 nm) with lower density; the bottom part is the interface layers close to SrTiO 3 substrate (~1.3 nm); and the middle part is the uniform R-LPCMO layers (~55 nm).
When fitting the O-LPCMO film, we leave the top two periods and bottom one period, and use the middle 50 periods to fit our superlattice strcture. Considering there may be intermixing layers, we use the interface layer with intermixing to fit our XRR data. From our fitting result, it is clear that the interfaces in the superlattice are really sharp and virtually have no intermixing (~ 0.01 nm).

Supplementary Note 4: Transport and magnetic properties of LCMO, R-LPCMO and O-LPCMO
We also measured the 40 nm pure LCMO film, as thick as the total thickness of LCMO in the O-LPCMO film as shown in Supplementary Figure 5. It shows that the MIT temperature of pure LCMO is about 50 K higher than that in the O-LPCMO. In addition, there is only one layer of PCMO between the LCMO layers, so we think the LCMO and PCMO are as a whole in O-LPCMO for charge circulates.

Supplementary Note 5: The imaging process and magnetic contrast inversion
The coexistence and competition of FMM and COI phases in the LPCMO system has been well known 7,8 . Although the easy magnetization axis is in-plane, an out-of-plane H field was applied upon MFM imaging. This leads to a perpendicular components of the magnetization, which is sufficient for the MFM to pick its contrast. Therefore, the no-magnetization area cannot come from in-plane magnetization. We assign these no-magnetization area as COI regions based on the well-known fact that the LPCMO system is featured by the coexistence and competition of the FMM and COI phases below T C , which has been extensively studied in the past 8,9 .
In order to subtract the morphology contribution from MFM signals, we perform the MFM imaging in the dual pass mode. The details of the imaging process can be found in our previous work 10 . From Fig. 3a and 3b in the main text, we can see clearly that the morphology can be perfectly removed from the MFM image with a 100 nm lift height and proper tuning of the feedback loop.
In this work, the 1 T field cooling was performed perpendicular to the sample surface so that the moments of the tip and the FM domains will be driven out of the plane and only the normal component of the ferromagnetic domain signals can be detected. The attractive force with negative force gradient caused by their interactions makes the cantilever effectively "softer", thereby reducing the resonant frequency of the cantilever and generate a negative phase shift at the resonance frequency [10][11][12] . Therefore, we could qualitatively interpret the MFM images in In order to clarify this issue, we also conducted MFM measurement by using the high coercivity (1.5 T~2 T) Co/Pt MFM tips 13 to pick up the magnetic contrast inversion, thus showing the zero phase signal areas are the AFM-CO states or the substrates. The sample and the tip were initialized under -9 T at 140 K. Then MFM images were acquired at -1 T, 1 T and 2 T to pick up the signal inversion. We get negative signals (attractive force between tip and sample) at -1 T, as shown in Supplementary Figure 6d and g. Since the coercivity of the sample is around 350 Oe at 140K, the 1 T field is large enough to overcome the coercivity of sample, but smaller than the coercivity of the tip. We get the positive signals (repulsive force between tip and sample) after at 1 T, as shown in Supplementary Figure  6e and h.
At last, we get negative signals again after going through the coercivity of the tip at 2 T, as shown in Supplementary Figure 6f and i. The signals show a clear inversion when the field going through the coercivity of the sample and the tip, and the nonmagnetic area still keep in no signal during the field changing, which prove the zero phase signal areas are the AFM-CO states or the substrates.

Supplementary Note 6: Comparison of the FMM volume fraction
We estimate the average FMM volume fraction from 5 images (20 μm × 20 μm × 5 images) at each temperature, and compare the FMM volume fraction from the SQUID initial magnetization curves, both for O-LPCMO and R-LPCMO samples under 1 T field cooling. The 1 T field cooling M-T curve and low temperature initial magnetization curves for both samples are also present for reference, as shown in Supplementary Figure 7a

Supplementary Note 7: Comparison of the FMM domain size
We did the domain size analysis from 5 images for both O-LPCMO and R-LPCMO samples at each temperature. The scanning region is 20 μm × 20 μm for each image. We compare the FMM domain size at the same T/T P rather than T/T C , because it is hard to determine the domain size after percolation (or below MIT temperature) when most domains join together. For this reason, it is not very meaningful to compare domain size at temperatures below MIT of either film (like 60 K). Before percolation (above MIT) for both films, the domain size of the O-LPCMO is clearly smaller than that of the R-LPCMO even at the same temperature (220 K), shown in Supplementary Figure 8. Therefore, conclusion that the O-LPCMO has smaller EPS domain size is firm no matter how one compares the two systems.