Theoretical / Experimental Results

RKKY - Interaction

The spin glass state differs intrinsically from normal magnetic systems. Famous examples of metallic spin glasses are copper with manganese (CuxMn1−x) and iron with gold (Au₁₋xF ex). An often examined, isolating spin glass is (EuS), which is magnetically thinned with unmagnetic (Sr)-ions (EuxSr1−xS); EuS itself is ferromagnetic. But there are two competing interactions: a negative cou-pling betweenadjacent Eu-ions and a positive bond between the atoms, which are

one row further away. Beside the temperature the concentration xis responsible for the magnetic behaviour.

Figure 2.2: Magnetic phase diagram of EuxSr1−xS

Figure 2.2 shows aphase diagramwith a direct transition from the magnetic phase to the spin glass phase at a concentration x of the Eu²⁺-ions between 13% and 51 %. Depending on the concentration x and the temperature T there exists a ferromagnetic (FM), a paramagnetic (PM) and a spin glass phase (SG).

For x-values between 51 and 65 % at first there is a transition from the para-magnetic phase to the ferropara-magnetic phase, when the temperature is lowered.

Because of the competing interaction the ferromagnetic order is highly disturbed;

but the spin glass phase exists only for deep temperatures [Ko93].

A theoretical explanation for the positive and the negative bonds is given by the RKKY-interaction, named after Rudermann, Kittel, Kasuya and Yosida. This exchange interaction is based on the ”polarisation” of the conduction electrons.

Each charged particle has a magnetic moment (spin) and so does the electron.

The polarised electrons itself influence the magnetic moments of the atoms, and so there is an intercation between the atoms themselves. For the strength of the bondJij it is valid:

Jij ∝ cos(2k~F ·r~ij)

r_ij³ (2.1)

There k~F is the Fermi-wavevector. For positive values of Jij(r) the interac-tion is ferromagnetic; negative values cause a negative interacinterac-tion. The

RKKY-interaction reaches many atoms in the neighbourhood and shows oscillatory be-haviour. Depending on the distance of the atoms there is a ferromagnetic or an anti-ferromagnetic bond of the spins (Fig. 2.3, left side). Because of the com-peting interaction and a statistical distribution of the atoms in the crystal there are spin glass effects. The atoms can be imagined in the middle of a concen-tric sphere with decreasing strength of interaction further away. From shell to shell of the sphere, ferromagnetic and antiferromagnetic behaviour alternates. A spin glass can develop, when electrons and atoms interact. The electrons carry the interaction between the atoms, whose spins can turn up or down under the influence of the surrounding electrons.

Figure 2.3: Schematic plot of the RKKY-interaction (left); Tag of four atoms (right)

Frustration

Approximately one half of the atomic pairs interacts ferromagnetic, the other half anti-ferromagnetic. This dual behaviour makes it possible that the spin of one atom cannot satisfy the interactions with all other atoms.

For illustration one can imagine a tag of four atoms which have an equal distance to one another (Fig. 2.3 on the right side). The interactions have the same amount, but for each pair of atoms the interaction is positive or negative.

For an odd number of positive (negative) couplings in the tag not all interactions can be satisfied at the same time. Every configuration of spins at least cannot satisfy one of the bonds; the system is frustrated. This frustration effect causes that there are several low energy states and thus different configurations of spins with the same minimum energy. One speaks of degenerated energy states. Such effects are also characteristic for combinatorial optimisation problems: the costs

are going to be interpreted as energy and thus one gets several equal energy states.

Phase Transition

Because of the degeneracy of the lowest energy states it can be asked, whether the spin glass is a new state of matter or just a very slow paramagnet. At a real phase transition the final state has a characteristic order as long as the temperature does not change. The spin glass phase could be a clearly distinct phase, whose magnetic order remains at low temperatures. But the spin glass could also be a paramagnet with a very slow magnetic behaviour; thus it just seems to be a statistical phase. If it could be observed that one or several spins change their orientation at low temperatures, then this would be a proof for paramagnetic behaviour. For this the spin glass must be observed over a very long period of time.

Susceptibility, Heat Capacity, Magnetisation

In the lab one can search for hints of a phase transition. Those hints would be sud-den changes in the magnetic and thermal characteristics at a critical temperature.

A lot of experiments show the spin glass phase, for example measurements of the alternating field capacity. Susceptibility χac gives information about the reaction of the spinsystem on a very weak, external alternating magnetic field.

Figure 2.4: Alternating magnetic field susceptibility of EuxSr1−xS

Figure 2.4 shows that χac has a sharp peak at the freezing temperature

Tf. But this peak is rounded off even for small additional fields; moreover it depends on the frequency and the concentration of the used materials. So spin glasses have a peak in susceptibility χ at a temperature Tf and that indicates a phase transition. The heat capacity C on the contrary has a wide maximum at a temperature higher than Tf. So what happens at the temperature Tf ?

Figure 2.5: Heat capacity and susceptibility for different magnetising forces At first, a phase transition into an antiferromagnetic order was assumed, but a suddenly appearing order should have shown up in the heat capacity. But this contradicts the fact that the specific heat is strictly monotonic increasing at Tf

and has a wide maximum foremost above Tf. Furthermore scattering experi-ments with neutrons show that there is no periodic order. Neither a homogenous magnetisation nor a antiferromagnetic structure can be observed.

Another important characteristic is the influence of the observation time in the freezing temperature of the spin glasses. If EuxSr1−xS is observed over a long period of time, Tf can change up to 20 percent. This shows that spin glasses do not come to a rest. There is a great spectrum of relaxation times, from the microscopic time of 10⁻¹² s to the time a spin needs to twist and up to many years. This behaviour can also be found in other incoherent systems like glasses, polymers and ceramics. Below Tf are many more or less equivalent spin configurations. The experimental realisation determines the taken states.

In order to understand the slow reaction of spin glasses at fields or other disturbances the magnetisation is measured. For the thermal equilibrium the mean magnetisation is M = 0. If the sample is cooled down without a magnetic field (zero field cooling) and then the external field is turned on for a short time, the sample remains magnetised (IRM). The same happens, when the sample is cooled down in a magnetic field (field cooling); after cooling the field is turned

Figure 2.6: Remanent magnetisation of an AuFe-alloy (left) and a computer simulation(right)

off. The magnetisation fades away very slowly. This remanent magnetisation depends on the previously applied field, the temperature, the switch-on time and the rate of cooling; its existence shows that there are many stable states in a spin glass. That is the main difference between incoherent materials and pure crystals.

The remanent magentisation is shown in Figure 2.6 on the left; the computer simulation on the right confirms a good synchronisation between experiment and theoretical model.

Review

The above listed phenomena can be understood with the frustration effects in a spin glass. Most of the materials with spin glass behaviour show two decisive effects: disorderand competition of the positive and negative couplings. This causes frustration and a high energy degeneracy of the system. In order to under-stand the characteristics of spin glasses, simplified models have been developed, which concentrate on the main mechanisms. In this way one gets a strongly idealised picture of a spin glass, which nevertheless contains all decisive physical aspects.