La2CuO4 is one of the High Tc superconducting cuprates

In its ground state, La2CuO4 has the configurations :

O2- : 2p6 non magnetic (mgn moment=0)

La3+ :4d10 5p6 ((mgn moment=0)

Cu2+ :4s0 3d9 (or 4s1 3d8  !!) (magn moment 0.3 to 0.5mB/Cu2+), while the free Cu+ should have 1.14mB/Cu2+.. The reduction of magnetic maoment is due to AF superexchange via Cu-O bridges

The stoechiometric compound is antifferomagnetic with Néel temperature TN=300K. 

The AF is complex (with 4 sublattices) because of the occurence of a weak ferromagnetic couplings and Dzyaloshinskii-Moriya anisotropy.

In the oxygen doped compounds, there is an O2- excess. These unpaired electons in the free radicals can give rise to paramagnetism.

Thus, in your compounds, all mentionned scenarios are possible : AF and paramagnetism of Cu2+, paramagnetism of unparied electrons of oxygen. Your M-T curve must be a superposition of several contributions.

La2CuO4 is one of the High Tc superconducting cuprates

In its ground state, La2CuO4 has the configurations :

O2- : 2p6 non magnetic (mgn moment=0)

La3+ :4d10 5p6 ((mgn moment=0)

Cu2+ :4s0 3d9 (or 4s1 3d8  !!) (magn moment 0.3 to 0.5mB/Cu2+), while the free Cu+ should have 1.14mB/Cu2+.. The reduction of magnetic maoment is due to AF superexchange via Cu-O bridges

The stoechiometric compound is antifferomagnetic with Néel temperature TN=300K. 

The AF is complex (with 4 sublattices) because of the occurence of a weak ferromagnetic couplings and Dzyaloshinskii-Moriya anisotropy.

In the oxygen doped compounds, there is an O2- excess. These unpaired electons in the free radicals can give rise to paramagnetism.

Thus, in your compounds, all mentionned scenarios are possible : AF and paramagnetism of Cu2+, paramagnetism of unparied electrons of oxygen. Your M-T curve must be a superposition of several contributions.

Subtract the two curves M(doped)-M(non doped) to get the contribution caused by doping. In addition, speculating the post annealing contribution curve does not look to be related directly to doping with magnetic oxygen, but rather a structural change affecting the magnetism.

The abrupt tiny peak ~ 35 K could be a structural phase transition, which gives the possibility that the other contributions happening smoothly above that temperature up to room temperature could be related to gradual structural transition (i.e. the structure is changing with temperature).

Here is a paper regarding thermal instability of La2CuO4 and it indicates the thermal lattice instability at 35 K observed at your annealled sample:

((Thermal-expansion-derived lattice instability in La2CuO4))

https://link.springer.com/article/10.1007/BF01307233

Therefore, most likely annealing the sample has had the major effect (not only the doped oxygen), and in order to verify the thermal dependence of the structure and separate it from the oxygen contribution then you could anneal a raw sample in Argon and re-measure the Magnetization vs. T and see if it is the same as the one annealed in oxygen or no. Thermal instability and dependence of the structure looks to be the major issue in La2CuO4.

Has the sample lattice structure degraded by annealing? Most likely yes

This link may be can help you:

https://link.springer.com/article/10.1007/BF01307233

@Dr. Mohammad El Hadifi: Thank you so much for the explanation. I got the idea why it is more to paramagnetic now.

@Dr. Mohammed Ebdah and @Dr. Yassine Slimani:

Thank you for the suggesting paper, it is answering my questions. I also annealed the sample in Ar condition and the graph is also same. Btw, how about the particle size effect. On the picture above in my question, the particle size of the sample is around 286 nm. If i measure a same sample with particle size 69 nm in oxygen condition at the same time and condition, I don't get any fluctuated paramagnetism (see picture), only pure paramagnetism. I wonder why no structural phase transition occur.  I annealed both the sample at the same time.

@Dr. Artur Braun: Thanks for the idea Dr. Now I don't have any knowledge about Hopkinson. I will write it down and learn about Hopkinson. 

Dear Collegue,

Your curve seems to be the contribution of two coexisting components: one paramagnetic (or superparamagnetic if there is hysteresis between ZFC and FC curves) and one ferromagnetic. I suggest to fit the paramagnetic part of the curve (low temperature) with a Langevin function, then sustraction of the experimental curve with the paramagnetic fitted curve will give you the ferromagnetic component. From the fit you will also can obtain the mean magnetic moment.

In this moment I don't know how to explain this spontaneous ferromagnetic component, that will be a very interesting subject to investigate.

Dear Faisal,

In my view, it would appear like there's a structural transition at low temperature in your sample. I suggest you measure in the reverse direction(cooling/warming) to confirm this if hysteresis occurs and also a field warming measurement in addition to the FC and ZFC will confirm whether a structural transition is taking place.

@Dr. P. Grima Gallardo: Thanks for the suggestion for fitting Dr. I will write down it.

@Dr. Zipporaj Wanjiku Muthui: Thank you for the suggestion Dr. I'll try measuring it in ZFC and FC condition. 

https://www.researchgate.net/publication/252541295_Investigation_of_the_phase_separation_in_superconducting_regions_of_La2CuO4d

Article Investigation of the phase separation in superconducting reg...

@Dr. Vaavilova. Thank you for sharing your work. I will carefully read it.