Open access peer-reviewed chapter

# Electronic, Transport and Magnetic Properties of Cr-based Chalcogenide Spinels

Written By

Chuan-Chuan Gu, Xu-Liang Chen and Zhao-Rong Yang

Submitted: May 30th, 2016 Reviewed: September 30th, 2016 Published: March 8th, 2017

DOI: 10.5772/66087

From the Edited Volume

## Magnetic Spinels

Edited by Mohindar Singh Seehra

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## Abstract

The Cr-based chalcogenide spinels with general formula ACr2X4 (A = Cd, Zn, Hg, Fe; X = S, Se) host rich physical properties due to coexistence of frustration as well as strong coupling among spin, charge, orbital and lattice degrees of freedom. In this chapter, recent advances on the study of electronic transport and magnetic properties of ACr2X4 are reviewed. After a short introduction of the crystal structure and magnetic interactions, we focus on the colossal magnetoresistance (CMR) in FeCr2S4, colossal magnetocapacitance (CMC) in CdCr2S4, negative thermal expansion (NTE) in ZnCr2Se4 and complex orbital states in FeCr2S4. It is hoped that this chapter will be beneficial for the readers to explore the interplay among different degrees of freedom in the frustrated system.

### Keywords

• Cr-based spinel
• colossal magnetoresistance
• colossal magnetocapacitance
• negative thermal expansion
• frustrated magnet
• complex orbital states

## 1. Introduction

Chromium-based chalcogenide spinels with general formula ACr2X4 (A = Cd, Zn, Hg, Fe; X = S, Se) have attracted special attention not only for pure academic interest, but also for potential applications [14]. On the one hand, the chromium-based chalcogenide spinels are typical strongly correlated electron materials. Due to strong coupling among spin, charge, orbital and lattice degrees of freedom, this system displays rich physical properties such as colossal magnetoresistance (CMR), colossal magnetocapacitance (CMC), gigantic Kerr rotation, and magnetoelectric effect [58]. On the other hand, the existence of frustration (magnetic or orbital frustration) increases complexity of the system, yielding complex behaviors such as spin glass, orbital glasses, and spin nematic [911]. Depending on external perturbation from magnetic field, electric field, chemical substitution, or disorder, different quantum states are subtle balanced, increasing difficulty for understanding the intrinsic physics of this system. For example, the polycrystalline (PC) sample of FeCr2S4 displays orbital ordering around 9K, while single crystal (SC) sample shows orbital glass [12, 13]. CdCr2S4 single crystal exhibits multiferroic behavior with the evidence of relaxor ferroelectricity and CMC [6]. However, the emergence of ferroelectricity and CMC effect in these thio-spinels was found to be highly sensitive to the detail of sample preparation and chemical doping. Both multiferroicity and CMC effect are absent in PC samples [14].

In this chapter, recent progresses in the studies of Cr-based chalcogenide spinels are described. We will focus on the origin of several physical effects including CMR, CMC and NTE. The origin of complex spin and orbital states will also be covered.

## 2. Crystal structure and magnetic interactions in ACr2X4 (A= Cd, Zn, Hg, Fe; X = S, Se)

The chromium-based spinels ACr2X4 (A = Cd, Zn, Hg, Fe; X = S, Se) have the cubic symmetry with space group Fd3m(Oh7), as is shown in Figure 1. Each cubic cell has 8 molecular formulae, which possess 56 ions, including 8 A2+ ions, 16 Cr3+ ions and 32 X2− ions. The X2− anions form a close-packed face-centered cubic lattice with two-type coordinated interstice: tetrahedral and octahedral ones. In the normal type spinel, the A2+ ions occupy 1/8 of the tetrahedrally coordinated interstices and the B3+ions occupy 1/2 of the octahedrally coordinated interstices. Otherwise, it is defined as an inverse spinel type, in which the A2+ and B3+ cations exchange the sites. In some situations, the compound with the same chemical elements could be formed by normal and inverse type spinel simultaneously [15, 16].

In ACr2X4, if magnetic cations (such as Mn2+, Co2+, Fe2+) occupy the A site, the system usually displays a ferrimagnetic order with a net magnetization, due to the opposite spin direction of A and B sites. On the other hand, if A site is occupied by nonmagnetic ions (e.g., Zn2+, Cd2+, Hg2+), the Cr3+ ions form a network of corner-sharing tetrahedral known as the pyrochlore lattice [17]. The dominant frustration in this network that hinders a simple antiparallel spin pattern is tightly associated with the way the Cr-Cr atoms are separated. The different ground states as a function of lattice constants are driven by the dominating exchange interactions: at small Cr-Cr separation, strong direct antiferromagnetic (AFM) exchange dominates, indicative of strong geometrical frustration [18]; at larger Cr-Cr separations, 90° Cr3+-X2−-Cr3+ ferromagnetic and Cr3+-X2−-A2+-X2−-Cr3+ or AFM Cr3+-X2−-X2−-Cr3+ superexchange interactions come into play [1], with additional bond frustration dominanting in the sulfides and selenides.

## 3. Colossal magnetoresistance in FeCr2S4

The presence of the CMR effect in mixed valence manganese oxides R3+1−xA2+xMnO3 (R is a rare earth and A is a divalent alkaline earth) in the vicinity of a paramagnetic (PM) to ferromagnetic (FM) transition has been the subject of much recent interest [19, 20]. This conductive behavior is described by the double-exchange (DE) model [21], which suggests that the conductivity is decided by the hopping of the electrons between hetrovalent Mn3+/Mn4+ pairs [22]. Later, another CMR material—FeCr2S4 has been reported by Ramirez et al. [5]. Since the magnetotransport behaviors of FeCr2S4 are quite similar to manganite perovskites, one might expect to apply the DE theory and Jahn-Teller (JT) polaron mechanism to explain the CMR effects in FeCr2S4. However, the X-ray diffraction (XRD) pattern and Mössbauer spectrum give direct evidences that there is no DE mechanism in FeCr2S4 [23].

The magnetization (M) in the temperature range from 4.2 to 400 K at the field H= 0.01 T is shown in Figure 2(a). With decreasing temperature, Mfirst increases slowly, and then increases abruptly near TConset(173 K) due to the PM-FM transition. The Curie temperature TC is 168 K, which is defined as the temperature corresponding to maximum of |dM/dT|. To clarify the magnetism in FeCr2S4, the temperature dependence of the PM susceptibility χis shown in Figure 2(b) as a plot of 1/χversus temperature. According to the physics of ferrimagnetism, the temperature dependence of the PM susceptibility χcan be described by Eq. (1) [25, 26]:

1/χ=T/C+1/χ0σ/(Tθ).E1

Here χ0,C, σand θare fitting parameters. The fit results of χare shown as the solid line in Figure 2(b), with 1/χ0 = 130(±2), C= 1.50(±0.02), σ= 3818(±50) and θ= 167(±3). Clearly, Eq. (1) fits the experimental result well except near TC, indicating a ferrimagnet state. At temperatures near and above TC, the system has strong short-range magnetic correlation, thus not in an ideal PM state.

The temperature-dependent resistivity ρand magnetoresistance MRH= [ρ(0) – ρ(H)]/ρ(0) from 4.2 K to 300 K under three different applied fields (0, 3 and 5 T) are shown in Figure 3. Both ρand MRH display a peak near TConset. The maximum of MR5 is 16%. The temperature corresponding to the minimum resistivity in the ρ-Tcurve below TC is defined as T1 = 153 K. As seen from the inset of Figure 3, the curve of ln ρversus 1000/Tis linear at both T>TConsetand T< T1, indicating a semiconductor like behavior in two regions. The fits by ρ= ρ0 exp(E/kBT) give the activation energies EL = 26 meV for T< T1, and EH = 47 meV for T>TConset, respectively. This implies that the conduction mechanisms are different for the two temperature regions. EH > EL indicates clearly that EH, the activated energy at temperatures above TConset, does not arise from a simple thermally activated effect. In order to understand the discrepancy between EL and EH, the thermoelectric power (S) is fitted by the experimental data through S= (kB/e)(a+ ES/kBT) [24, 27]. ESis obtained as 23 meV, which is approximately equal to 26 meV for EL, and much lower than 47 meV for EH. The discrepancy between ESand EH in manganite perovskites can be attributed to the presence of lattice or magnetic polarons. Since there is neither a structure transition nor a JT effect around TC in FeCr2S4, the polarons mainly exist in the form of magnetic polarons [28].

The sample at temperatures above TConsetis in a typical PM state, in favor of the existence of magnetic polarons [24]. As is known, a magnetic polaron is a carrier coupled by short-range magnetic correlation within a magnetic cluster at temperatures above TConset. Thus, the effective mass of a magnetic polaron increases greatly with respect to that of a naked carrier, which means it has a lower mobility and a higher activated energy [29, 30]. In an ideal FM order, magnetic polarons will be delocalized from the self-trapped state and turn into naked carriers completely [29]. In consequence, ρdecreases drastically at this point, accompanied with the system changing completely to one of thermal-activated transport of naked carriers. However, ρdecreases gradually from TConsetto T1, and MR also expands to a broader temperature range below TConset, indicative of the existence of magnetic polarons in this temperature range. Magnetic polarons will be delocalized gradually as the PM phase weakens. They then vanish completely at T1, denoting that magnetic order destroys the environment for magnetic polarons. Hence, ρin the temperature range from TConsetto T1 can be described by a two-fluid model involving the coexistence of magnetic polarons and naked carriers. It could also explain that MR expands to a broader temperature range below TConset, since the formation of magnetic polarons is inhibited by the enhancement of FM order at an external magnetic field.

To further investigate the evolution of magnetic polarons in FeCr2S4, nonmagnetic Cd is substituted for Fe. The XRD measurement for Fe1−xCdxCr2S4 reveals that the substitution of Fe by Cd produces no structural change [31]. The resistivity ρand magnetoresistance MRH= [ρ(0) – ρ(H)]/ρ(0), as a function of temperature are shown in Figure 4. Comparing with FeCr2S4, the Cd-containing sample has a higher zero-field resistivity. For x= 0.2, a peak is shown in both ρand MRHnear TC. Zero-field ρreveals semiconductor-like behavior at temperatures for both T> TC and Tfar below TC, which is similar to the results of FeCr2S4. Upon substitution of Cd for Fe, both the zero-field ρand MRHvalue increases monotonically, while the peak of the MRHcurve shifts to lower temperatures. As mentioned above, it is believed that the conduction originates from Fe2+ narrow band at temperatures far below TC, and is dominated by the magnetic polarons in the temperature region above TC. For Fe1−xCdxCr2S4, there are two sided influences of the introduced Cd2+ on the conduction behavior. On the one hand, since the conduction originates from Fe2+ narrow band, the increase of Cd ions will decrease carrier density, resulting in the increase of ρ. As is mentioned above, the magnetic polaron hopping conduction ρ= ρ0Texp(E/kBT) should be considered for T> TC. On the other hand, the introduced Cd ions will induce both Coulomb and magnetic potential fluctuations, as a sequence, leading to the formation of a mobility edge. Thus, thermally activated behavior ρ= ρ0 exp(E/kBT) due to a mobility edge may be more appropriate to describe the conduction above TC [32, 33].

Figure 5 shows the zero-field resistivity curves replotted as ln ρ −1000/Tand ln(ρ/T) 1000/T, respectively. For x= 0.2, the curve of ln ρ −1000/Tis linear both at T> TC and at temperatures far below TC. The activation energy E, which is obtained from the fits to ρ= ρ0 exp(E/kBT), is 39 meV for the temperatures far below TC, and 50 meV for TC. As is known, the discrepancy of the activation energy for FeCr2S4 is described by a two-fluid model concerning the coexistence of magnetic polarons and naked carriers. As is shown in Figure 5(a), magnetic polaron hopping conduction [ρ= ρ0Texp(E/kBT)] is more appropriate for the resistivity behavior at T> TC. Moreover, it gives more convincible description as compared to thermal-activated mechanism for higher Cd concentration, which implies that the magnetic polarons may be more stable in the high-Cd samples.

Direct evidence of the magnetic polarons could be best sought with electron spin-resonance spectroscopy (ESR) measurements [31]. Above T* ~ 113 K, each spectrum consists of a single line with a Lorentzian line shape, and further decreasing temperature, some distortions occur. A similar behavior has been observed in manganese perovskites, which is attributed to the presence of the FM clusters embedded in PM matrix [34, 35]. In addition, as Tis increased, the peak-to-peak line-width ΔHPPdecreases in a broad temperature range [31]. Tmin (corresponding to the temperature where ΔHPPis minimum) is much higher than T*, suggesting that the relaxation mechanism dominating in CdCr2S4 is different with that in the manganese. The analysis of ESR experiments in CMR materials has pointed out that line-width ΔHPPwill decrease exponentially upon warming if the magnetic polaron acts as a part in the relaxation mechanism, as a result of the motionally narrowed relaxation mechanism [36]. Thus in CdCr2S4, since magnetic polaron is more stable upon Cd substitution, it is plausible that the hopping of magnetic polarons will make contribution to the relaxation process, and certainly decrease the line-width ΔHPP. The decrease of line-width with increasing temperature obviously gives a direct evidence for the presence of magnetic polarons in CdCr2S4.

For a pure magnetic polaron system, the magnetic polaron forms by self-trapping in a ferromagnetically aligned cluster of spins at T< TP(TPdenotes the temperature where polarons start forming). As the temperature is lowered toward TC, the polaron grows in size. According to the ML model [37], for a low carrier density ferromagnet, when ≈ 1 is accomplished, magnetic polarons overlap, which leads to the carrier delocalization. Here nis the carrier density, andξis FM correlation length. It provides the temperature ratio of about T/TC ≤ 1.05–1.1 [37]. Therefore, for a pure magnetic polaron system, the magnetic polarons are periodically spaced before overlapping, similar to a gas of magnetic polarons. When the system is in a PM state, no magnetic correlation occurs among the polarons to be considered as superparamagnetic. On the other hand, if the system is inhomogeneous, there may be magnetic correlation, which means magnetic polarons themselves are no longer isolated. Due to the correlation, the system could not stay in the superparamagnetic state, which results in the existence of some distortions in the resonance line [34]. Hence, it can be concluded that magnetic polarons first exist as a gas of polarons at temperatures T< TP, and then transform into correlated polarons at temperatures T< T* (TP> T* > TC).

## 4. Colossal magnetocapacitance in CdCr2S4

Multiferroic materials that exhibit simultaneous magnetic and ferroelectric order as well as concomitant magnetoelectric coupling have attracted special interest in recent years [3840]. CdCr2S4 was originally investigated as a FM semiconductor more than 40 years ago [41]. Recently, it was reported that SC CdCr2S4 exhibits multiferroic behavior with the evidence of relaxor ferroelectricity and CMC [6]. Soon after, similar magnetoelectric effect has also been revealed in other related thio-spinel compounds, that is, CdCr2Se4 and HgCr2S4 [42, 43]. However, the emergence of ferroelectricity and CMC effect in these thio-spinels was found to be highly sensitive to the detail of sample preparation and chemical doping. Annealing SC samples in vacuum or sulfur atmosphere led to a suppression of relaxation features, and no remanent electric polarization could be found at low temperatures [14]. In addition, multiferroicity and CMC effect are absent in undoped PC samples, but present in indium-doped PCs [14]. Accordingly, the question of whether the magnetoelectric effect in CdCr2S4 is intrinsic or not has been raised [44, 45]. In this section, the origin of CMC effect in the CdCr2S4system would be revealed systematically through the magnetic, dielectric and electric transport measurements of CdCr2S4 and Cd0.97In0.03Cr2S4 PC samples before and after annealing in cadmium vapor.

Upon doping and annealing, no impurity phases are detected [46]. The M-Tcurves under an external magnetic field of 1 T for all samples are shown in the inset of Figures 6(b) and 7(b). After annealing in cadmium vapor, the magnetic properties of PC CdCr2S4 and Cd0.97In0.03Cr2S4 are slightly changed. The Curie temperature TC is about 92.1 K for CdCr2S4 and 92.9 K for Cd0.97In0.03Cr2S4, respectively. After annealing, the increase of TC is less than 4 K. Figures 6 and 7 display the temperature dependence of dielectric constant ε′and ac-conductivity σ′at two different frequencies in magnetic field of 0 T and 4.5 T for the as-prepared and annealed samples. Both ε′ and σ′ increase with temperature monotonously and do not show anomaly and magnetic field dependence in the whole measured temperature range, as seen from Figures 6(a, b) and 7(a, b). After annealing CdCr2S4 at 800°C (Figure 6(c, d)), ε′ displays a hump in the low temperature region and the hump temperature increases with frequency, implying a relaxor-like behavior of the dielectric property [14]. However, the magnetic field of 4.5 T still has no obvious effect on either ε′or σ′.

In contrast, for the Cd0.97In0.03Cr2S4 sample annealed at 380°C (Figure 7(c, d)), a strong upturn of ε′ is clearly observed with decreasing temperature near TC. σ′ has a similar temperature dependence as ε′. The external magnetic field of 4.5 T does not change the shape of ε′-Tand σ′-Tcurves, but makes the upturn of ε′ and σ′ shifting toward higher temperatures. Magnetocapacitance, defined as MC = (ε′(4.5 T) − ε′(0 T))/ε′(0 T), reaches up to 290% at 1 kHz and 1950% at 600 kHz (Figure 8(a)). The sample annealed at 800°C has similar temperature dependence of ε′ and σ′ as the sample annealed at 380°C, while has a much higher ε′ and σ′, and a lower MC value (reaches up to 145% at 1 kHz and 104% at 600 kHz, see Figure 8(c)). A clear feature in Figures 6(c, d), and 7(c, f) is that the appearance of CMC effect is accompanied by the field-enhanced ac conductivity. The inset of Figure 6(d) shows the temperature dependence of DC-resistivity under 0 T and 4.5 T for the annealed CdCr2S4 sample. In accordance with the field independence of the ac-conductivity, the application of 4.5 T magnetic field has no evident influence on the DC-resistivity. Upon cooling, the DC-resistivity increases monotonously as a typical semiconductor behavior in the whole temperature range measured. However, as seen in Figure 8(b, d), the temperature dependence of DC-resistivity for the annealed Cd0.97In0.03Cr2S4 samples is quite different from that of CdCr2S4. For annealed Cd0.97In0.03Cr2S4, the zero-field resistivity first increases with decreasing temperature. After reaching up to a maximum near TC, the resistivity decreases abruptly, indicating the occurrence of insulator-metal transition. Upon further cooling to the low temperature region, the resistivity increases again. Being correlated to the insulator-metal transition around TC, the external magnetic field of 4.5 T makes the resistivity peak moving to a higher temperature and dramatically depresses the peak value. Magnetoresistance, defined as MR = (ρ(0 T) − ρ(4.5 T))/ρ(0 T), reaches up to about 95% for the sample annealed at 380°C and 93% for the sample annealed at 800°C, much higher than the value of the most-investigated CMR material FeCr2S4, see the insets of Figure 8(b, d) [5, 31].

The appearance of CMC in the CdCr2S4 system is always accompanied by CMR. To confirm that the CMC can be described by a combination of CMR and Maxwell-Wagner effects, the impedance spectroscopy of Cd0.97In0.03Cr2S4 annealed at 380°C is fitted to a Maxwell-Wagner equivalent circuit. The impedance spectroscopy is converted from capacitance and dielectric loss tangent in the frequency range from 100 Hz to 600 kHz. The circuit consists of two subcircuits in series, each containing a resistor (R) and a capacitor (C) in parallel, as depicted in the inset of Figure 9(a). Zis better described by using the Cole-Cole equation: Z= R/(1+(iωRC)n), (0 ≤ n≤ 1) [47, 48]. The fitting was performed with all parameters unlocked. Take the data under zero magnetic field for the Cd0.97In0.03Cr2S4 sample annealed at 380°C for example. The impedance spectroscopy can be fitted well in the temperature range from 79 K to 189 K, see the solid lines in Figure 9(a). The temperature-dependent total resistance R1 + R2 derived from the fitting coincides with the temperature dependence of the measured DC resistance, as shown in Figure 9(b).

The perfect match between the resistance derived from the impedance spectroscopy and the DC resistance strongly suggests that the annealed Cd0.97In0.03Cr2S4sample contains two components with different electrical response. These two components can be a sample and an electrode, or a grain and a grain boundary. In these cases, an applied bias voltage can change the capacitance and the ac conductance [5052]. Moreover, Hemberger et al. have experimentally excluded any electrode effects on the magnetodielectric response and excluded any nonhomogeneous impurity distribution in their SC CdCr2S4 [42, 45]. Thus, CMC effect could not be attributed to the extrinsic Maxwell-Wagner effect. In phase-separated manganite, strong magnetodielectric effect is resulted from the scenario of phase separation [5356]. The thermoelectric-power and ESR measurements reveal the existence of magnetic polarons in the annealed Cd0.97In0.03Cr2S4 sample [49]. As is discussed in the above section, magnetic polaron is similar as the presence of the FM clusters embedded in PM matrix, indeed a kind of phase separation in nature [24, 57, 58]. For the exchange interaction with localized spins, the energy of the conduction electron in the FM cluster is smaller than that in the PM host, which results in a nanoscale charge and phase separation [57, 58]. Upon cooling, magnetic polarons grow in size, as a consequence, their overlap induces larger metallic clusters and finally a percolative metallic filament. After the insulator-metal transition, the coexistence of FM metallic and PM insulating phases is replaced by a complete FM order founded in all the regions. The interfacial polarization or space-charge yielded by the mixture of insulating and metallic regions would cause a dielectric response, which is intrinsic to the material. Therefore, the dielectric response can be attributed to a combination of magnetoresistance and an intrinsic Maxwell-Wagner effect, as observed in the phase-separated manganite [5356].

## 5. Negative thermal expansion in ZnCr2Se4

Recently, ZnCr2Se4 has been observed to display NTE and a large magnetostriction [2]. To study the very origin of NTE, a set of experimental techniques is utilized to probe the spin-lattice correlation in ZnCr2Se4. Figure 10(a) shows the temperature dependence of lattice constant a. With decreasing temperature, afirst decreases rapidly and then manifests a NTE behavior below about TE = 60 K. Upon further cooling below 20 K, splitting of several peaks in XRD spectra is observed. Figure 10(b) presents the magnetization versus Tat low-applied magnetic field of 100 Oe. A fitting of the inverse susceptibility 1/χaccording to the PM Curie-Weiss (CW) law 1/χ= (T– ΘCW0)/Cis shown in Figure 10(c). A large positive CW temperature ΘCW0 = 85 K and the coefficient C= 3.74 are obtained. The large positive CW temperature implies a dominant FM exchange interaction; however, the compound shows an AFM ordering at low temperatures. On the other hand, both the constant g-factor and linear behavior of peak-to-peak line-width (ΔHPP) in the ESR spectrum evidence a well-defined PM state at least above 100 K [59]. Thus, the inverse susceptibility should be described by the PM CW law down to this temperature. However, it departs from the linear behavior at a temperature as high as about 180 K, see Figure 10(c). Recalling the fitting process in Figure 10(c), a constant CW temperature ΘCW0 has been assumed, that is, a constant magnetic exchange interaction J. In addition, the nearest neighbor FM Cr-Se-Cr and other neighbor AFM Cr-Se-Zn-Se-Cr super-exchange interactions depend strongly on the lattice constant [1]. It means that the total Jmay change since a decreases dramatically upon cooling. Accordingly, the traditional CW behavior should be modified within the present case. In specific, one should take a variable ΘCW (or J) as a function of Tor ainto account.

In AFM spinel oxides, the CW temperature changes exponentially with the lattice parameter [60]. Naturally, an empirical description of ΘCW(T) = ΘCW0 Θ*eT/βis postulated. The fitting of the inverse susceptibility above 100 K using the modified CW behavior 1/χ= (T– ΘCW(T))/Cis exhibited in Figure 11(a). The parameters are Θ= 226 and β= 45. Furthermore, a remarkable deviation below 100 K in blue short-dashed line indicates the appearance of the effective internal field originating from FM clusters. Next, based on the obtained Θand β, ΘCW(T) is extrapolated to low temperatures as exhibited in Figure 11(b). It shows a derivation at about 180 K from the nearly constant value. With further lowering temperature, it decreases faster and faster and below T≈ 45 K, ΘCW(T) even becomes negative. These features may interpret qualitatively the fact that ZnCr2Se4 is dominated by ferromagnetic exchange interaction but orders at low temperatures antiferromagnetically. Since the exchange integral and the CW temperature are linked by J(T) ΘCW(T) [J(a) ΘCW(a)], we will use J(T) [J(a)] instead in the following discussion. Given that Jis changeable, magnetic exchange and lattice elastic energies can link effectively with each other via magnetoelastic coupling. The free energy Fin a magnetoelastic system is expressed as:

F(T)=J(T)ijSiSj+12Nω2Δ2(T)TS(T)E2

where Nis the number of the ion sites, ωis the averaged vibrational angular frequency, and Δis the averaged strain relative to the equilibrium lattice constant. The first term is exchange energy (Eex) as a function of J, the second is lattice elastic energy (Eel) related mainly to the lattice parameter a(or equivalent T), and the last is the entropy. From the above equation, if the system stays at an ideal PM state, then Eex = 0. So Eex and Eel will show no coupling.

When the system stays in FM state with a changeable J, there may exist a competition between Eex and Eel since the former is negative while the latter always is positive. Indeed, it has been concluded above that Jdecreases exponentially [Figure 11(a)] and some FM clusters form gradually below 100 K. Therefore, the concomitant decrease of Jand acauses an increasing Eex but decreasing Eel in the FM clusters. At some critical point, a totally compensation between them may present. If a further decreases upon cooling, the variation of Eex would gradually exceed that of Eel in magnitude. Especially when Jdrops sharply with respect to a, say here at TE, a tiny decrement of awill give rise to a dramatic increment of Eex, whereas Eel keeps nearly constant. The state in a system subjected to stimuli, such as cooling, always tends to develop toward one that can lower F. In this sense, it is favorable to lowering Fby expanding the lattice parameter ato increase J(J> 0), and thereby to decrease Eex due to its negative value, at the same time at a cost of small increases of Eel in magnitude. This means that a NTE of the lattice originating from the FM clusters with an exponentially changeable Jis expected. It is noted that when applying a magnetic field to the system in the NTE temperature region, magnitude of NTE enhances [2]. This is because the size or population of the FM clusters increases when applying a magnetic field. On the contrary, when an AFM ordering appears, Jbecomes negative and the condition to stimulate NTE is not met any more. Normal thermal expansion upon cooling results in a simultaneous decreasing of Eex and Eel, which is consistent to lowering F. In fact, a normal expansion feature is observed below TN [2]. It should be noted that the existence of NTE evidences in turn that Jis changeable [2, 61]. If Jkeeps constant in a FM cluster, Eex will be almost constant and a normal thermal expansion of the lattice alone can give rise to a decrease of Eel and of Fsufficiently.

The thermal expansion data for ZnCr2(Se1−xSx)4 (0 ≤ x≤ 0.1) SC samples is represented as the function of δ= [L(T) − L0]/L0 from 4 K to 300 K in Figure 12(a), where L0 is the length of the sample at room temperature. For ZnCr2Se4, as Tis lowered, δfirst decreases monotonically and reaches a minimum at Tm~ 50 K. Then δbegins to increase, followed by a steep downturn upon further cooling across TN. For the S-doped samples, the striction below TN becomes gentler, while the temperature region of the NTE is expanded. As seen from the inset of Figure 12(a), TN moves to lower temperatures with increasing S content while Tm shifts oppositely. To further probe the nature of spin-lattice coupling in the complex magnetic state, the field dependence of both magnetization and magnetostriction δat 5 K for all the ZnCr2(Se1−xSx)4 (0 ≤ x≤ 0.1) samples are illustrated in Figure 12(b, c). δis defined as δ= [L(H) − L(0)]/L(0), and L(0) is the length of the sample at zero field. Due to strong spin-lattice coupling, a one-to-one correlation between M-Hand δ-His clearly reflected in Figure 12(b, c). At both of the characteristic fields of HC1 and HC2, δalso displays an anomaly. Being associated to domain reorientation, hysteresis is also observed below HC1 as the magnetic field returns to zero, see Figure 12(b, c). For x= 0.05 and 0.10, field dependence of magnetization is still inconsistent with the magnetostriction data. With increasing S content, the hysteresis behavior below HC1 wanes in both M-Hand δ-H, and completely vanishes at x= 0.10 [9]. In addition, the magnetization and magnetostriction data are both suppressed by an order of magnitude.

To investigate the NTE and magnetostriction in S-substituted samples, the inverse susceptibility was analyzed by the same method as is proposed above, which is shown in Figure 12(d) [59]. For ZnCr2Se4, ΘCW (T) drops dramatically and changes sign at 43 K, close to the onset temperature of NTE. The substitution of S for Se drives ΘCW0 to lower temperature on account of an enhancement of the AFM exchange interaction, while the temperature corresponding to the sign change in ΘCW (T) is increased. It could explain why the temperature region of the NTE is extended in ZnCr2(Se1−xSx)4 (0 < x≤ 0.1) samples. The substitution of sulfur for selenium not only enhances the AFM exchange interaction, but also leads to a Cr-Cr bond disorder [9]. The bond disorder frustrates the spins and finally leads to a spin-glass state. In a spin-glass state, since the spin arranges or freezes randomly upon cooling to lower temperature, the magnetic exchange and lattice elastic energy are decoupled. Note that, in the S-substituted samples, potential striction or shrinkage of the lattice can still be observed upon cooling across TN, which implies that a partial long-range AFM ordering still manifests and coexists with the spin-glass state.

## 6. Complex orbital states in spinel FeCr2S4

Due to strong frustration, the spinel FeCr2S4 not only shows the fascinating CMR effect, but also displays complex orbital states [12, 13]. Recently, it was reported to display the orbital glass state in SC samples, while it displays orbital ordering in PC ones [12, 13]. In FeCr2S4, the Fe2+ ion is tetrahedrally coordinated by the sulfur ions, with an electronic configuration eg3t2g3and S= 2, is a typical JT active. Thus, it might induce long-range orbital order at low temperatures [62, 63]. In addition, the diamond lattice formed by Fe ions is geometrically frustrated for the orbital degrees of freedom [12, 13]. Thus, Fichtl et al. proposed two possible explanations to interpret the discrepancy of different orbital states in FeCr2S4 [12]. One is that the orbital glass state is attributed to geometric frustration in the SC, and broken in the PC by marginal disorder. Alternatively, the orbital ordering in the SC is suppressed by small disorder (such as chlorine defects from the growth process). To verify the role of disorder in these two possibilities, the physical properties of some SC and doped PC FeCr2S4 are presented in detail next.

Figures 13(a) and 14 display the magnetization as a function of temperature obtained in ZFC and FC processes with H= 0.005 T. All samples display clear anomalies around 65–75 K and irreversibility between ZFC and FC curves. The cusp-like anomaly was early correlated to an abrupt increase of magnetic anisotropy below Tm ~ 70 K, and recently attributed to a spin-reorientation transition associated with the onset of short-range orbital ordering [64, 65]. Moreover, the PC FeCr2S4 displays a step-like anomaly in the magnetization around TOO ~ 9 K in Figure13(a). For the other samples, the transition is replaced by a smooth temperature dependence, which is shown in the insets of Figure 14. The step-like anomaly for PC and smooth temperature dependence for other ones can be related to the orbital-ordering (OO) transition and orbital glass state, respectively [66]. A similar temperature-dependent magnetization between SC and doped PC samples suggests that the orbital order might also be frozen in these samples. Temperature dependence of specific heat for PC FeCr2S4 is shown in Figure 13(b). It displays a well-defined λ-type anomaly around 9 K, indicative of the OO transition, which is consistent with the step-like transition in magnetization [12, 13]. For the other samples, the λ-type anomaly is entirely inhibited and becomes a broad hump, see Figure 14(e). In addition, as Tis lowered below 2 K, the heat capacity approaches zero following a strict T2-dependence in the insets of Figure 14(e). In contrast, the specific heat of PC deviates from the linear behavior in low temperatures below 2 K, shown in the inset of Figure 13(b). The earlier reports pointed out that the phonon and magnon contribution to the specific heat of SC could be ignorable as compared to the orbital contribution at ultra-low temperatures [12, 13]. Thus, the existence of the orbital glass state in both SC and doped samples is plausible, since heat capacity of orbital order suppressed by random fields obeys the T2-dependence predicted by the theoretical calculation [67]. Clearly, all the results reveal that the orbital moment has been frozen into the orbital glass state in the SC and doped PC samples.

For a different orbital state, the orientation of the orbital is coupled to the elastic response of the ionic lattice via electron-phonon interaction, and therefore a different orbital state is accompanied by a different charge distribution [68]. The resistivity curves in the function of ln ρ-1000/Tand ln ρ-(1000/T)1/4 are shown in Figures 13(c) and 15, respectively. The conduction mechanism for PC FeCr2S4 has been studied in detail in the previous section. In the low temperatures below 37 K, the resistivity could be better described by a semiconductor-like behavior. Upon warming, a curvature emerges, which might be related to the formation of short-range orbital order state [12]. For the other samples, the Mott’s variable-range hopping is more appropriate to fit the feature of resistivity. As we know, for variable-range hopping, it is necessary that the system exists random potential fluctuations, being related to the degrees of disorder [32]. Hence, the presence of random potential could be the reason for the emergence of orbital glass state [12, 13].

In conclusion, complex orbital states in FeCr2S4 are driven by the coexistence of strong electron-phonon coupling and geometrical frustration. The different orbital states in PC and SC FeCr2S4 reveal the exquisite balance between frustration and strong coupling among different degrees of freedom. The magnetism, heat capacity and resistivity properties of all the samples give clear evidence that the disorder in SC and doped samples raises random potential up, freezes the orbital order and finally results in orbital glass state.

## 7. Conclusion

Due to the presence of frustration as well as strong coupling among spin, charge, orbital and lattice degrees of freedom, Cr-based chalcogenide spinels display rich physical effects and complex ground states. There is still some open questions such as the nature of spin nematic, correlation between orbital state and magnetic structure, which should be addressed in future under the conditions of high pressures and high magnetic fields.

## Acknowledgments

This work is financially supported by the National Natural Science Foundation of China under Grant nos. U1332143 and 11574323.

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Written By

Chuan-Chuan Gu, Xu-Liang Chen and Zhao-Rong Yang

Submitted: May 30th, 2016 Reviewed: September 30th, 2016 Published: March 8th, 2017