The pseudocubic lattice parameter of the film is 3.85 ˚A

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PHY401 Exercise Sheet 13

HS2020 Prof. Dr. Johan Chang

Due on December 14 Discussion on December 15

Exercise 1 SQUID data of La₂NiMnO₆

A 5.4 nm-thick double-perovskite La2NiMnO6 thin film has been grown coherently strained on a (001)- oriented diamagnetic SrTiO₃ substrate. Determine the magnetic moment (inµ_B/f.u.) of the film from the magnetization versus field data acquired with a SQUID magnetometer. (Hint: Assume a linear background). Explain the origin of the measured magnetization and how it compares with the expected nominal value.

Figure 1 shows the sample used for the measurement. The pseudocubic lattice parameter of the film is 3.85 ˚A. The data file can be downloaded on the course website.

Figure 1: The La2NiMnO6 thin film on the SrTiO3 substrate. The scale of the paper is 1 mm.

Exercise 2 Width of the diffraction maximum

We assume that in a linear crystal on every lattice point ρ~ = m~a, m ∈ Z, there is an identical point-like scattering centre. The total amplitude of the scattered radiation is proportional to F = Pexp

−im~a·∆~k

. Using the series expansion

M−1

X

m=0

x^m = 1−x^M

1−x (1)

we find for the sum over M lattice points

F =

1−exp

−iM~a·∆~k

1−exp

−i~a·∆~k

. (2)

(a) The scattered intensity is proportional to|F|². Show that

|F|²≡F^∗F = sin²

1

2M~a·∆~k

sin² 1

2~a·∆~k

. (3)

(b) Plot the function in equation 3 for a few different values of M in a suitable range ofa·∆k. You will see that there is a main peak with some side peaks left and right. Show that the width of the main peakw (from minimum to minimum) is given byw= _aM^π .

1

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Figure 2: X-ray diffraction data for a series of NdNiO₃ films. The used wavelength was λ= 1.5406 ˚A.

(a) Four identical films grown on different substrates (LAO, NGO, STO, KTO). (b) Films of varying thicknesses grown on NGO.Source: Scherwitzl et al., Adv. Mater. 22, 5517 (2010)

(c) Figure 2 shows X-ray diffraction results on NdNiO₃ films in the vicinity of the (001) Bragg reflection. Besides the reflection from the substrate (the very sharp peak) you can see an intensity distribution as expected from equation 3 (thewiggles). Use the data in panel a) to determine the effective c axis lattice parameter of the film for the four different substrates.

(d) Using the results of exercises b) and c), determine the film thickness for the samples corresponding to the yellow and red curves, marked by arrows in panel b) of figure 2.

Hint: You will have to translate the x-axis units to units of ∆~k. Consider the following sketch, where|~ki| ≈ |~kf|:

~ki

~k_f

∆~k 2θ

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