Magnetic, optical, and magnetooptical properties of spinel-type ACr 2X4 (A = Mn, Fe, Co, Cu, Zn, Cd; X = O, S, Se)

Journal of the Physical Society of Japan

Vol. 77, No. 3, March, 2008, 034713

#2008 The Physical Society of Japan

Magnetic, Optical, and Magnetooptical Properties

of Spinel-Type ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, Cd; X ¼ O, S, Se)

ꢀ

y

1

2

Kenya OHGUSHI , Yoichi OKIMOTO , Takeshi OGASAWARA ,

z

1

1;2;3

Shigeki MIYASAKA , and Yoshinori TOKURA

¹Department of Applied Physics, University of Tokyo, Tokyo 113-8656

²Correlated Electron Research Center (CERC), National Institute of Advanced Industrial

Science and Technology (AIST), Tsukuba, Ibaraki 305-8562

³Multiferroics Project, ERATO, Japan Science and Technology Agency (JST),

c/o Department of Applied Physics, University of Tokyo, Tokyo 113-8656

(Received November 14, 2007; accepted December 28, 2007; published March 10, 2008)

A comprehensive study of magnetic, optical, and magnetooptical properties was carried out for single

crystals of the spinel-type ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd; X ¼ O, S, and Se). The optical

reﬂectivity measurements for 0.1–30 eV revealed a wide variation in electronic structures on a large

energy scale between oxides (X ¼ O) and chalcogenides (X ¼ S and Se). For A ¼ Fe and Co, we

observed the intra-atomic d–d transitions of A^2þions with a tetrahedral coordination, and successfully

deduced the crystal ﬁeld splitting ꢀE, the Racah parameter B, and the spin–orbit coupling constant ꢀ by

analysis based on the ligand ﬁeld theory. A comparison of these optical parameters between oxides and

chalcogenides indicated the strong covalency eﬀect in the chalcogenides. In A ¼ Cu, the insulator–metal

transition between X ¼ O and Se was clearly demonstrated by optical conductivity spectra. Magnetic

properties were discussed in relation to electronic structures. A compound with a small optical gap is

typically a ferrimagnet with antiparallel arrangements of A^2þand Cr^3þspins, whereas a compound with a

large optical gap undergoes ﬁrst-order phase transition into spiral spin ordering at a low temperature. We

found that the magnetic anisotropy constants K₁for ACr₂S₄(A ¼ Mn, Fe, and Co) are approximately

scaled by the inverse of the intra-atomic d–d transition energies of A^2þions in agreement with the

second-order perturbation theory for single-ion anisotropy. The magnetooptical spectra in a wide energy

range (0.2–4.5 eV) were measured for chalcogenides focusing on the d–d transition resonance. We

4

observed gigantic magnetooptical signals up to 4.1^ꢁin the energy range of A₂! T₂and A₂! T₁

transitions of Co^2þions for CoCr₂S₄, and analyzed them in the framework of the ligand ﬁeld theory. We

propose that the strong covalency of the ligand sulfur, as well as the local breakdown of inversion

symmetry, in the tetrahedral site plays a crucial role in the enhancement of magnetooptical responses.

KEYWORDS: optical spectroscopy, chromite, spinel chalcogenide, ferrimagnetism, magnetooptical Kerr effect,

d–d transition

DOI: 10.1143/JPSJ.77.034713

These compounds undergo a magnetic phase transition at

the critical temperature (T_c) into either a canted state (Yafet–

1. Introduction

´

Among a number of spinel-type compounds denoted Kittel structure) or a collinear Neel state with reduced

AB₂X₄, the most famous series is the B ¼ Cr system, in magnetic moments, as characterized by the magnetic

which Cr^3þions occupy the octahedral site. Mn, Fe, Co, Ni, contribution to fundamental Bragg diﬀractions. Bertaut and

Cu, Zn, Cd, and Hg are incorporated in the tetrahedral A site Durac performed a representation analysis of these states

usually with a nominal valence of þ2. A remarkable aspect in NiCr₂O₄.¹⁸⁾When A ¼ Mn, Fe, and Co,¹⁹⁾an additional

of the spinel structure is that the X site accommodates magnetic transition occurs at T⁰(< T_c), below which a

various elements of group VI (i.e., O, S, Se, and Te). conical spin structure (spiral spin state with uniform mag-

Physical properties vary widely from oxides (X ¼ O) to netization) with a modulation vector of ꢀ ¼ ꢂꢁ (110) is

chalcogenides (X ¼ S, Se, and Te) and hence they have realized.^5–11)According to the mean-ﬁeld theory of the

often been investigated separately.

classical Heisenberg model, a conical spin state is stabilized

Research on oxide spinels with B ¼ Cr (chromite) started by a strong competition between antiferromagnetic A–B

in 1950s as a natural extension of research on ferrite (Fe- and B–B interactions.²⁰⁾Unexpectedly, there is an addi-

containing spinel-type oxides).¹⁾The magnetic properties of tional magnetic transition at T⁰⁰(< T⁰) in CoCr₂O₄, which

compounds with magnetic A^2þions (A ¼ Mn, Fe, Co, Ni, is identiﬁed as a lock-in transition from an incommensu-

and Cu) have been investigated by magnetization,²⁾neutron rate phase to a commensurate phase (ꢁ ¼ 2=3).^9,11)When

diﬀraction,^3–13)electron spin resonance (ESR),^14,15)and A ¼ Fe, Ni, and Cu, there also remains an orbital degree

nuclear magnetic resonance (NMR)^16,17)measurements. of freedom on A^2þions. Structural transitions from cubic

symmetry to tetragonal symmetry in FeCr₂O₄(T_JT¼ 140 K,

^ꢀPresent address: Institute for Solid State Physics, University of Tokyo,

c=a < 1),²¹⁾NiCr₂O₄(T_JT¼ 320 K, c=a > 1),^22,23)and

Kashiwa, Chiba 277-8581.

^yPresent address: Department of Chemistry and Materials Science, Tokyo

Institute of Technology, Meguro-ku, Tokyo 152-8551.

^zPresent address: Department of Physics, Osaka University, Osaka 560-

0043.

CuCr₂O₄(T_JT¼ 850 K, c=a < 1),^23–25)where T_JTis the

transition temperature and c=a is the ratio of the tetragonal

lattice constants, have been attributed to cooperative Jahn–

Teller (J–T) transformations associated with tetrahedrally

034713-1

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

coordinated A^2þions. These issues have also been exten- of FeCr₂S₄near 10 K has been ascribed to the orbital-glass

sively investigated from a theoretical viewpoint.^26–29)Quite transition associated with the two-fold degenerated orbitals

recently, CoCr₂O₄has been reported to exhibit ferroelectric of Fe^2þions.^104–106)

polarization, which can be reversed by an external magnetic

In this paper, we report a systematic study of spinel-type

ﬁeld, below T⁰.³⁰⁾Chromite with nonmagnetic A^2þions compounds denoted ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and

(A ¼ Zn, Cd, and Hg) is an ideal system for searching Cd; X ¼ O, S, and Se) using high-quality single crystals

interesting phenomena related to geometrical magnetic grown by the chemical vapor transport method and the

frustration, because Cr^3þspins (S ¼ 3=2) are located on a ﬂux method. By supplementing the magnetic and optical/

corner-sharing network of tetrahedra (pyrochlore lattice). magnetooptical properties with a new set of systematic

The spin correlation in ZnCr₂O₄, which can be monitored experimental data over a wide energy region, we present a

not only by inelastic neutron scattering but also by optical uniﬁed view of the electronic structures and their relevance

phonon spectroscopy,³¹⁾develops as large spin clusters to magnetic properties for both oxides and chalcogenides.

(hexagonal loops) far above T_c,³²⁾culminating in a spin- Such a study has been expected to be the basis for under-

Peierls-like phase transition at 13 K.³³⁾In ACr₂O₄(A ¼ Cd standing other intriguing phenomena including magnetic,

and Hg), a half-magnetization plateau was observed under a elastic, dielectric, and their mixed properties. The format of

high magnetic ﬁeld.^34,35)

this paper is as follows: In §2, we describe details of the

On the other hand, Cr-based chalcogenide spinels with sample preparation and measurement procedures. In §3, we

nonmagnetic A^2þions (A ¼ Zn, Cd, and Hg) have been present the optical conductivity spectra of all the compounds

widely known as magnetic semiconductors, where magnetic investigated and we clarify the covalency eﬀect on the basis

Cr^3þspins are viewed as naturally introduced into II–VI of the strong-coupling ligand ﬁeld theory. The obtained

semiconductors AX (ZnS, ZnSe, and so on).^36–46)The results are used in §4 to explain the material dependence

ordering patterns of Cr^3þspins range from a ferromagnetic of magnetism. §5 is devoted to the magnetooptical spectra

state in CdCr₂Se₄to a helical spin state (spiral spin state of chalcogenides and also to the elucidation why these

with no uniform magnetization) in ZnCr₂Se₄.^47–54)The compounds exhibit large magnetooptical responses. Part of

salient feature of these magnetic semiconductors appears the magnetooptical spectra of ACr₂S₄(A ¼ Fe and Mn) has

in their (magneto)optical properties.^55–69)For example, the been published in the form of a short paper.⁹⁶⁾The summary

red shift of the absorption edge upon cooling below the of this paper is given in §6.

magnetic transition temperature (T_c) in ZnCr₂Se₄, CdCr₂S₄,

and CdCr₂Se₄has raised the possibility of controlling

2. Experiment

magnetic instability by adjusting the band gap of the host 2.1 Sample preparation

semiconductors, and vice versa.^55–57,59)As another example,

All the crystals investigated in this study except

magnetooptical Kerr eﬀect spectroscopy has demonstrated CuCr₂O₄were grown by the chemical vapor transport

that CdCr₂S₄and CdCr₂Se₄show fairly large signals (Kerr method.^{76,107–110)}We ﬁrst synthesized polycrystalline sam-

rotation amounting to ꢃ0:5^ꢁ).^40,61,63,64)Recent research ples from the starting materials in powder form; they are

topics for these materials are the mutual coupling between MnO, FeO, CoO, and Cr₂O₃for the oxides, and Mn, Fe,

magnetic and electric properties such as the magnetoelectric Co, Cu, Zn, Cd, Cr, S, and Se for the chalcogenides. The

70)

eﬀect in ZnCr₂Se₄

and the relaxor ferroelectricity in stoichiometric amounts of the source materials were mixed,

CdCr₂S₄and HgCr₂S₄.⁷¹⁾The interplay between spin, ground, and sealed in an evacuated quartz tube with 11 mm

charge, and lattice degrees of freedom has been explored diameter and 200 mm length. The tube was heated for 24 h at

in Cr-based chalcogenide spinels with A ¼ Mn, Fe, Co, and 630 ^ꢁC for CuCr₂Se₄and at 900 ^ꢁC for the other compounds.

Cu; the former three compounds are ferrimagnetic insula- In synthesizing chalcogenides, we raised the furnace temper-

tors and the A ¼ Cu system is a ferromagnetic metal.^72–80)ature slowly between 300 and 400 ^ꢁC to avoid bursting the

Here, we should note that Cu atoms take a monovalent ampoule. The powdered MnCr₂O₄, FeCr₂O₄, and CoCr₂O₄

(not divalent) state in chalcogenides, as revealed by neut- were respectively green, brown, and blue, while CuCr₂O₄

ron diﬀraction analysis,⁸¹⁾photoemission spectroscopy,^82,83)and all the chalcogenides investigated were black. In

soft x-ray magnetic circular dichroism (MCD) measure- growing single crystals of ACr₂O₄(A ¼ Mn, Fe, and Co),

ments,^84,85)and band calculations;⁸⁶⁾this renders the system ACr₂S₄(A ¼ Mn, Fe, and Co), and ACr₂Se₄(A ¼ Zn

exceptionally metallic. The large negative magnetoresist- and Cd), we used CrCl₃as a transport agent.¹⁰⁷⁾2.0 g of

ance near T_chas been reported in Fe_1ꢄxCu_xCrS₄.^87–91)The polycrystalline samples and 0.04 g of CrCl₃were sealed in

anomalous Hall eﬀect (AHE) in Cu_1ꢄxZn_xCr₂Se₄has been an evacuated quartz ampoule with 20 mm diameter and

discussed in connection with their nontrivial magnetic 100 mm length. We placed the ampoule inside a three-zone

structures,^92,93)and AHE in CuCr₂Se_4ꢄxBr_xhas been furnace of 950 ^ꢁC at the center with a temperature gradient

reported to occur without dissipation.⁹⁴⁾The mechanisms of 10 ^ꢁC/cm for one month. The obtained single crystals

of gigantic magnetooptical Kerr eﬀect responses in FeCr₂S₄with typical dimensions of 2 ꢅ 2 ꢅ 2 mm³were octahedral

(4.3^ꢁ),^95–97)CoCr₂S₄(ꢃ4^ꢁ),^98–101)and CuCr₂Se₄(1.2^ꢁ)^102,103)characterized by (111) surfaces. In growing the single

have been elucidated to be an intra-atomic d–d transition crystals of CuCr₂Se₄, we used I₂as the transport agent.⁷⁶⁾

A

resonance in the ﬁrst two compounds and a plasma quartz ampoule of 20 mm diameter and 200 mm length

resonance in the last compound. Photoinduced spin dynam- containing 2.0 g of polycrystalline CuCr₂Se₄and 0.32 g of I₂

ics of ACr₂S₄(A ¼ Fe and Co) and ACr₂Se₄(A ¼ Cu and was kept in a furnace with a cold zone of 800 ^ꢁC and a hot

Cd) were well accounted for in terms of the magnetic zone of 950 ^ꢁC for 2 weeks. The thus-obtained crystal was

anisotropy constant.^95,97)The broad hump in the speciﬁc heat 5 ꢅ 5 ꢅ 0:2 mm³and plate-shaped.

034713-2

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

Table I. Electron conﬁgurations, lattice parameters (a and c), magnetic structures in ground state, and magnetic transition temperatures (T_c) of single-

ꢁ

crystalline ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd; X ¼ O, S, and Se). The space groups are I4₁=amd tetragonal symmetry for CuCr₂O₄and Fd3m

cubic symmetry for the other compounds. We classify here the magnetic structure of CuCr₂Se₄as ferromagnetic for historical reason; however, it can be

viewed as ferrimagnetic arrangements of Cr, and Cu and Se spins (see text for details). When ﬁrst-order transitions are present below T_c, their transition

temperatures (T⁰and T⁰⁰) are also listed. The T_cof CuCr₂Se₄is cited from the literature.^76Þ

Electron conﬁguration

Lattice constant

˚

(A)

T_c, T⁰, T⁰⁰

Compound

Magnetism

(K)

A site

Cr site

MnCr₂O₄

FeCr₂O₄

CoCr₂O₄

CuCr₂O₄

MnCr₂S₄

FeCr₂S₄

Mn^2þ: e²t₂³

Fe^2þ: e³t₂³

Co^2þ: e⁴t₂³

Cu^2þ: e⁴t₂⁵

Mn^2þ: e²t₂³

Fe^2þ: e³t₂³

Co^2þ: e⁴t₂³

Cu^1þ: e⁴t₂⁶

Zn^2þ: e⁴t₂⁶

Cd^2þ: e⁴t₂⁶

Cr^3þ: t₂³_g

a ¼ 8:439ð1Þ

Conical

48, 14

93, 35

97, 27, 12

152

a ¼ 8:367ð5Þ

Conical

a ¼ 8:337ð1Þ

Conical

a ¼ 6:038ð2Þ, c ¼ 7:784ð5Þ

a ¼ 10:113ð3Þ

a ¼ 10:003ð3Þ

a ¼ 9:928ð3Þ

Canted

Conical

67, 4

170

Ferrimagnetic

Ferromagnetic

Helical

CoCr₂S₄

CuCr₂Se₄

ZnCr₂Se₄

CdCr₂Se₄

228

2:5

2g

Cr^3:5þ: t

a ¼ 10:341ð2Þ

a ¼ 10:499ð4Þ

a ¼ 10:747ð3Þ

430

Cr^3þ: t₂³_g

22

Ferromagnetic

135

Single crystals of CuCr₂O₄with dimensions of 1 ꢅ 1 ꢅ 1 grating monochromator (0.6–5 eV), which are equipped

mm³were prepared by the ﬂux decomposition method.²⁴⁾with a microscope and a continuous helium ﬂow optical

11.7 g of K₂Cr₂O₇powder, 2.5 g of CuO powder, and 0.15 g cryostat. In order to measure the optical reﬂectivity spectra

of B₂O₃powder were weighed, mixed, and placed in a in the vacuum–ultraviolet range, we utilized the synchrotron

15 cm³Pt crucible with a cover. It was kept in ﬂowing air radiation at the Institute for Molecular Science (UVSOR)

inside the tube furnace, which was placed in a draft chamber. supplemented with a grating monochromator. We deduced

Temperature was raised up to 805 ^ꢁC at a rate of 100 ^ꢁC/h, the optical conductivity spectra from the reﬂectivity spectra

kept for 24 h, then slowly cooled to room temperature at a by Kramers–Kronig analysis, where we adopted !^ꢂ(ꢂ ¼

rate of 8 ^ꢁC/h. The residues were washed well with NaHSO₃3{5) extrapolation for reﬂectivity data above 30 eV.

solutions.

We performed magnetooptical Kerr spectroscopy in the

photon energy range of 0.2–4.5 eV, utilizing the polarization

modulation technique in the polar Kerr conﬁguration.¹¹¹⁾

2.2 Characterization and magnetic measurements

Powder X-ray diﬀraction analysis with the use of the Measurements were carried out for the as-grown (111)

Cu Kꢂ radiation for the crushed crystals conﬁrmed that, surfaces, because mechanical polishing turned out to enlarge

aside from CuCr₂O₄with the I4₁=amd tetragonal symmetry, the coercive force, making the reverse of magnetization

ꢁ

all the samples investigated here show the Fd3m cubic using a permanent magnet (supplying a magnetic ﬁeld of

symmetry at room temperature; their lattice parameters about 0.2 T) diﬃcult. The measurement of the magneto-

estimated are listed in Table I. The inductively coupled optical Kerr eﬀect in the energy range of 0.2–0.8 eV was

plasma analysis indicated that the ratios of A atoms to Cr enabled by applying circular dichroism spectroscopy.^112–115)

atoms are 1 : 2 within experimental accuracies. Resistivity The experimental setup is schematically drawn in Fig. 1.

was measured by the conventional four-probe technique The light incident from the Fourier transform infrared

with use of gold paste as the electrodes. Magnetization was (FT-IR) spectrometer was linearly polarized by a wire grid

measured using a superconducting quantum interface device polarizer. Then, polarization was modulated by the photo-

(SQUID) magnetometer. Magnetic torque measurements elastic modulator (Hinds, Model II/ZS50) at a frequency (f)

were performed with use of the commercial torque magne- of 50 kHz, where we ensure that the retardation amplitude

tometer (Quantum Design Physical Property Measurement does not exceed ꢃ=2 in all the energy range investigated.

System), where a piezoresistive cantilever was used. The The light reﬂected from the sample was analyzed by a wire

magnetic ﬁeld of 7 T was rotated within the (110) plane. The grid polarizer and probed by an MCT detector. The f and

raw data contain extra contributions probably from the 2f components of the signals were detected by a lock-in

gradient of the applied magnetic ﬁeld at the sample position; ampliﬁer, and returned to the FT-IR spectrometer. The thus-

this term was successfully subtracted so that the resultant obtained interferograms for the dc, f, and 2f components

data exhibit the correct angle dependence expected for the were Fourier-transformed with special care about the phase

crystal symmetry.

correction;¹¹⁶⁾the resultant waves of these components

basically correspond to the reﬂectivity, Kerr ellipticity, and

Kerr rotation spectra, respectively. The calibration of signals

2.3 Optical measurements

Optical reﬂectivity spectra were measured on the (111) was performed similarly to that in the literature,¹¹¹⁾where

surface in the photon energy (!) range of 0.1–30 eV. As- we used a sapphire plate operating as the ꢄ=4 plate at several

grown specular surfaces were used without further mechan- speciﬁc energies. In order to deduce the polarization change

ical polishing in chalcogenides and FeCr₂O₄, while the of magnetic origin, we carried out the same measurement

surfaces of ACr₂O₄(A ¼ Mn, Co, and Cu) were polished twice at each temperature — before and after reverting the

with alumina powders before optical measurements. We magnetic ﬁeld. The contribution of the Faraday rotation of

used a Fourier-transform interferometer (0.1–0.8 eV) and a the cryostat window was eliminated by subtracting the data

034713-3

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

FT-IR

A

B

C

A B

C

L

MnCr₂O₄

FeCr₂O₄

MnCr₂S₄

FeCr₂S₄

MCT

I(0)

I(f), I(2f)

P (φ = 45 deg)

A (φ = 0 deg)

PEM: ZnSe

( f = 50 kHz)

Out

Ref.

In

CoCr₂O₄

CuCr₂O₄

CoCr₂S₄

Lockin Amp.

Sample

H

0.2

0

Fig. 1. Experimental setup for magnetooptical Kerr spectroscopy in the

photon energy range of 0.2–0.8 eV. The light generated by a light source

(L) in the Fourier transform infrared (FT-IR) spectrometer is transmitted

through the polarizer (P) oriented at 45^ꢁ, modulated by the photoelastic

modulator (PEM) at a frequency (f) of 50 kHz, and irradiated on the

sample surface under a magnetic ﬁeld (H). The reﬂected light is detected

by the MCT detector after passing through an analyzer (A) at 0^ꢁ. The

lock-in detected f and 2f components of the signal, together with the dc

component, are input into the FT-IR spectrometer producing interfero-

grams for the dc, f, and 2f components.

CuCr₂Se₄

ZnCr₂Se₄

CdCr₂Se₄

0

10

20

30

Photon Energy (eV)

0.3

0

10

20

30

Photon Energy (eV)

above T_c. The technique described here is useful for

detecting the magnetooptical responses of mobile carriers

in metals.^117,118)For the experiment in the energy range of

0.7–4.5 eV, we constructed a setup similar to that in Fig. 1,

where we employed a grating monochromator, Glan–Taylor

polarizers, and a photoelastic modulator of Hinds, Model II/

CF57; spectroscopy in this photon energy range is more

conventional.¹¹¹⁾

Fig. 2. Room-temperature reﬂectivity spectra in photon energy range of

0–30 eV for (left panel) oxides ACr₂O₄(A ¼ Mn, Fe, Co, and Cu) and

(right panel) chalcogenides ACr₂S₄(A ¼ Mn, Fe, and Co) and ACr₂Se₄

(A ¼ Cu, Zn, and Cd). The optical processes associated with the

respective absorption structures are basically assigned as follows: O 2p,

S 3p and Se 4p ! A 3d and Cr 3d transitions in the A region; O 2p,

S 3p and Se 4p ! A 4s=4p and Cr 4s=4p transitions in the B region;

O 2s, S 3s and Se 4s ! A 3d and Cr 3d transitions in the C region.

Exceptional cases are ACr₂Se₄(A ¼ Cu, Zn, and Cd), where Se 4p !

A 4s=4p and 5s=5p transitions are shifted downwards to the A region.

3. Optical Properties

3.1 Overall feature of optical conductivity spectra

roughly estimated from Fig. 2: The S 3s and Se 4s orbitals

Figure 2 shows the optical reﬂectivity spectra in the wide are ꢃ9 eV higher than the O 2s orbital; the S 3p and Se 4p

photon energy range (0–30 eV). We ﬁrst identify the optical orbitals are ꢃ4 eV higher than the O 2p orbital. This

process in the oxides (X ¼ O) (left panel of Fig. 2). For this, observation is semiquantitatively consistent with the photo-

we refer to the assignment in the perovskite-type oxides emission spectroscopy for II–VI semiconductors^121,122)

denoted RMO₃(R and M being rare earth and 3d transition- and the band structure calculation for spinel-type com-

metal elements, respectively)¹¹⁹⁾and the spinel-type pounds.^86,123)The ﬂoating up of the S- and Se-related bands

Fe₃O₄.¹²⁰⁾Optical absorptions below 2.5 eV, which strongly for the large energy scale up to 4–9 eV markedly enhances

depend on A atoms, are discussed in the following sub- the covalency, and transfers the spectral weight in the optical

section. The transitions between 2.5 eV and the marked conductivity spectra to the lower energy range. This is one

plasma edge at around 11 eV (the A region) are assignable to of the origins of the considerable diﬀerence in the physical

the O 2p ! A 3d and Cr 3d charge transfer excitations; this properties between oxides and chalcogenides. The A region

transition consists of at least three broad bands, which are of the A ¼ Cu, Zn, and Cd systems shows considerably

likely caused by the crystal ﬁeld splitting of the 3d bands of rich spectral structures in comparison with that of the

the transitional-metal atoms. The spectral structures in the B A ¼ Mn, Fe, and Co systems. This most likely originates

region are attributable to the O 2p ! A 4s=4p and Cr 4s=4p from the overlapping of Se 4p ! Cu 4s=4p, Zn 4s=4p, and

transitions. The optical absorptions above 16 eV (the C Cd 5s=5p transitions;⁶⁸⁾these absorptions correspond to the

region) correspond to the O 2s ! A 3d and Cr 3d transi- interband transitions from the valence band to the con-

tions, which become dipole-allowed owing to the strong duction band in Cu₂Se and in the II–VI semiconductors

mixing of the A 3d and Cr 3d orbitals with the O 2p ZnSe and CdSe.

orbitals; internal structures in this region should be under-

stood in the same manner as in the A region.

Chalcogenides (X ¼ S and Se) (right panel of Fig. 2)

3.2 Intra-atomic d–d transition of Fe^2þand Co^2þions

Figure 3 shows the optical conductivity spectra of ACr₂X₄

exhibit similar spectral features to oxides while showing (A ¼ Mn, Fe, and Co; X ¼ O and S) in the low energy

a downward shift in energy. This lowering of the energy range. We ﬁrst focus on the oxides [Fig. 3(a)]. There are

scale is relevant to the upward shift of the S- and Se-related distinct spectral structures near and below the optical edge.

bands toward the Fermi level. The energy level diﬀerence is The broad hump structure at around 1.6 eV common to the

034713-4

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

former is a parity-odd dipole-allowed transition because of

5

⁵E_gꢅ T_1u¼ ⁵T_1uþ T_2u, whereas the latter is parity-even

(a)

(b)

4

and silent in reﬂectivity measurement. Similarly, the A₂!

⁴T₁transition of the Co^2þion exhibits Davydov splitting into

4

the dipole-allowed A_2g! T_2utransition (⁴A_2gꢅ T_2u

¼

MnCr₂O₄

MnCr₂S₄

⁴T_1u) and the dipole-forbidden A_2g! T_1gtransition. The

4

⁴A₂! T₂transition of the Co^2þion also splits into the

4

⁴A_2g! T_1uand A_2g! T_2gtransitions; nevertheless, the

4

FeCr₂O₄

FeCr₂S₄

parity-odd transition is not dipole-allowed, because A_2g

ꢅ

⁴T_1ucontains no ⁴T_1u; this transition is revived with the help

of phonon ﬁelds, which is beyond the scope of the present

study. To be exact, these transitions should be classiﬁed

200

2000

0

ꢁ

by the crystal symmetry O_h(m3m) after introducing the

Davydov splitting eﬀect. To avoid confusions, however, we

hereafter identify them by using a notation based on the local

CoCr₂O₄

CoCr₂S₄

ꢁ

symmetry of the A site, T_d(43m).

0

As seen in Fig. 3(a), the d–d transitions of the A^2þions

show ﬁne structures even at room temperature (far above

T_cand T_JT). They have their origin in spin–orbit coupling,

whose eﬀect is treated up to the ﬁrst order of the spin–

orbit coupling constant ꢀ. Below T_c, the spectra are also

expected to show exchange splitting, which is dealt with

as in the case of Zeeman splitting. The level scheme

including these two eﬀects is drawn in Fig. 4. In order

to calculate the oscillator strength of each transition, we

ﬁrst evaluate the matrix element of the electronic dipole

moment hSꢂMꢅjP_ꢂjS⁰ꢂ⁰M⁰ꢅ⁰i in the unit of the reduced

0

1

2

3

0

1

2

Photon Energy (eV)

Fig. 3. Optical conductivity (ꢆ) spectra in low energy range for (a) oxides

denoted ACr₂O₄(A ¼ Mn, Fe, and Co) and (b) chalcogenides denoted

ACr₂S₄(A ¼ Mn, Fe, and Co). Open triangles indicate absorptions due to

the intra-atomic d–d transition of the Cr^3þion, A_2gðt₂³_gÞ ! T_2gðt₂²_ge_g¹Þ.

4

Closed triangles represent the intra-atomic d–d transitions of the A^2þ

ions: ⁵Eðe³t₂³Þ ! T₂ðe²t₂⁴Þ for A ¼ Fe, and ⁴A₂ðe⁴t₂³Þ ! T₂ðe³t₂⁴Þ and

5

4

⁴A₂ðe⁴t₂³Þ ! T₁ðe³t₂⁴Þ with increasing energy for A ¼ Co.

4

three spectra (marked with open triangles) is also observed matrix element hSꢂjPjS⁰ꢂ⁰i with the use of Wigner–Eckart’s

in CdCr₂S₄and CdIn₂S₄:Cr,¹²⁴⁾being ascribable to the theorem

59)

intra-atomic d–d transition of the Cr^3þion in the octahe-

hSꢂMꢅjP_ꢂjS⁰ꢂ⁰M⁰ꢅ⁰i

dral environment, A_2gðt₂³_gÞ ! T_2gðt₂²_ge_g¹Þ. Meanwhile, A-

dependent absorptions with sharp peak structures (marked

with closed triangles) are attributable to the intra-atomic d–d

4

1

0

pﬃﬃﬃﬃﬃﬃ

¼

hSꢂjPjS ꢂ ihꢂꢅjꢂ ꢅ T_1uꢂiꢁðSS ÞꢁðMM Þ;

ð1Þ

ðꢂÞ

transitions of the A^2þions, which are tetrahedrally coordi- where P is the electronic dipole moment, jSꢂMꢅi the spin-M

nated by the ligand O^2ꢄions with the T_d(43m) site and orbital-ꢅ wave function of the ^2Sþ1ꢂ multiplet term,

ꢁ

symmetry. We analyze these excitations on the basis of the ðꢂÞ the dimension of the irreducible representation ꢂ,

ligand ﬁeld theory by strong-coupling approximation.^125–128)hꢂꢅjꢂ⁰ꢅ⁰T_1uꢂi the Clebsch–Gordan coeﬃcient, and ꢁ the

Kronecker delta symbol. Then, we obtain oscillator strength,

2

We note here that neither the strong- nor weak-coupling

scheme is appropriate for the present system, because the

which is proportional to jhSꢂMꢅjP_ꢂjS⁰ꢂ⁰M⁰ꢅ⁰ij . The calcu-

crystal ﬁeld splitting between the e and t₂orbitals ꢀE lations were performed under two conditions, where the

is actually comparable with the Racah parameter B. No magnetic moments are aligned to the [001] axis alone

spin-allowed d–d transition is expected for Mn^2þions (single-domain case), or otherwise are equally aligned along

in accordance with our experimental result. The 0.5 eV one of the [100], [010], and [001] axes (multidomain case).

absorption in FeCr₂O₄is assignable to the spin-allowed d–d In both cases, the light k-vector is assumed to be normal to

transition, Eðe³t₂³Þ ! T₂ðe²t₂⁴Þ, with a transition energy of the (001) surface for simplicity. The calculated results are

5

summarized in Fig. 4 for the ⁵E ! T₂transition of the

! ¼ ꢀE ; this excitation is also observed in MgAl₂O₄:Fe,¹²⁹⁾

4

Fe^2þion, and for the A₂! T₁transition of the Co^2þion;

ZnS:Fe,^129,130)CdIn₂S₄:Fe,¹³¹⁾and CdTe:Fe.¹²⁹⁾Theoretical-

4

ly, there are three spin-allowed d–d transitions in Co^2þthe oscillator strength for the A₂! T₂transition of the

ions. However, the ⁴A₂ðe⁴t₂³Þ ! T₁ðe²t₂⁵Þ transition is a two- Co^2þion becomes zero in current formalism (see discussion

4

particle excitation with a rather small oscillator strength. in preceding paragraph).

Hence the experimentally observed absorptions at around

4

To gain insight into the d–d transition, we ﬁt the opti-

0.8 and 2.0 eV are assigned to the ⁴A₂ðe⁴t₂³Þ ! T₂ðe³t₂⁴Þ cal conductivity [ꢆð!Þ] spectra with the multicomponent

4

transition with ! ¼ ꢀE and the A₂ðe⁴t₂³Þ ! T₁ðe³t₂⁴Þ tran- Lorentz oscillator model

sition with ! ¼ ꢀE þ 12B, respectively.¹³²⁾

S_j!²!_jꢅ_j

X

ꢆ ¼

;

ð2Þ

The intra-atomic d–d transitions in ordinary compounds

2

ð!_j²ꢄ !²þ ꢅ_j²Þ þ 4!²ꢅ_j²

j

are parity-even, thereby dipole-forbidden. In the present

system, however, since the A^2þions occupy two sites where S_j, !_j, and ꢅ_jare the oscillator strength, the transition

without local inversion symmetry in the primitive unit cell, energy, and the damping constant for the j-th oscillator,

the d–d transitions become dipole-allowed through Davydov respectively. Since the present materials are paramagnetic at

splitting.^133,134)The E ! T₂transition of the Fe^2þion is room temperature, we adopt the oscillator strength calcu-

5

split into the E_g! T_1uand E_g! T_2gtransitions; the lated in the multidomain case without any exchange split-

034713-5

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

Fig. 4. (a) Local environment of A atom in ACr₂X₄.

The A atom is tetrahedrally coordinated with X

atoms. (b), (c) Schematic energy level diagrams

obtained using the ligand ﬁeld theory for the Fe^2þ

and Co^2þions in a tetrahedral environment. The

eﬀects of crystal ﬁeld with T_dsymmetry, spin–

orbit (L–S) coupling, and exchange interaction are

introduced step by step from the left to the right.

Only the energy levels responsible for spin-allowed

one-particle excitations are drawn. The oscillator

(a)

(c) Co²⁺

⁴T₁

X

A

Γ₆

Γ₈

5

ζ

12

1

6

(b) Fe²⁺

Γ₇+Γ₈

1

4

−

ζ

Γ₁+Γ₄+Γ₅

Γ₇

ΔΕ+12Β

1

ζ

2

5

ζ

12

⁵T₂

Γ₈

5

strengths of the ⁵Eðe³t₂³Þ ! T₂ðe²t₂⁴Þ transition

1

⁴T₂

Γ₃+Γ₄

1

6

4

of the Fe^2þion and the ⁴A₂ðe⁴t₂³Þ ! T₁ðe³t₂⁴Þ

−

ζ

4

Γ₆+Γ₈

transition of the Co^2þion are calculated under two

conditions: the single-domain case in which all the

spins are aligned to the [001] direction and the

multidomain case in which equal numbers of [100],

[010], and [001] magnetic domains are present. The

ratio of oscillator strength is shown in the ﬁgure; the

ﬁlled, hatched, and open arrows respectively repre-

sent the ꢀM_z¼ þ1; 0; ꢄ1 transitions (M_zbeing the

magnetic quantum number along the z-direction).

Γ₅

−

1

−

ζ

ΔΕ

3

4

ΔΕ

+ + – + + 0 + – 0 +

+ – – – + 0 – 0 – –

Γ₁+Γ₂+Γ₃+Γ₄+Γ₅

⁵E

⁴A₂

Γ₈

9

15

9

5

10

4

10 1 4

5

1

10 15

1

5

4

6

5

Single

Multi

Single

Multi

T_d

L-S

Exchange

T_d

L-S

Exchange

5

ting. The model for the E ! T₂transition of the Fe^2þion

consists of three-component oscillators with !₁¼ ꢀE ꢄ

ð3=4Þꢀ, !₂¼ ꢀE ꢄ ð1=4Þꢀ, !₃¼ ꢀE þ ð1=2Þꢀ, S₁¼ 3S,

S₂¼ 5S, and S₃¼ 7S; hence, there are six ﬁtting parameters:

60

40

20

0

300

(a)

FeCr₂O₄

(b)

FeCr₂S₄

200

100

0

4

ꢀE, ꢀ, S, ꢅ₁, ꢅ₂, and ꢅ₃. To ﬁt the A₂! T₁transition of

the Co^2þion, we adopt the three-component model with

!₁¼ ꢀE þ 12B ꢄ ð1=4Þꢀ, !₂¼ ꢀE þ 12B þ ð1=6Þꢀ, !₃¼

ꢀE þ 12B þ ð5=12Þꢀ, S₁¼ 3S, S₂¼ 2S, and S₃¼ S. More-

over, we introduce two Lorentz oscillators to express the

0.4

0.6

0.2

0.4

0.6

Photon Energy (eV)

⁴A_2g! T_2gtransition of the Cr^3þion and the Cr charge

4

transfer excitation in the A range in Fig. 2. Since the nature

60

40

20

0

400

200

0

of the A₂! T₂transition of the Co^2þion is complicated

from a microscopic point of view, we here make the

assumption that the number of orbital degeneracies has a

one-to-one correspondence to oscillator strength; then we

4

(c)

CoCr₂O₄

(d)

CoCr₂S₄

4

acquire the ﬁtting model identical to that of the A₂! T₁

transition of the Co^2þion with a change in excitation energy.

As shown in Figs. 5(a), 5(c), and 5(e), a better ﬁtting is

achieved by these ﬁtting procedures. The obtained optical

parameters are summarized in Table II.

0.6

0.8

1

0.6

0.8

1

Photon Energy (eV)

The ꢀE values for the Fe^2þand Co^2þions are 0.488 eV

and 0.843 eV, respectively, being much smaller than those

for an octahedral environment (typically 1.5–2.5 eV).^125–128)

This reﬂects the small number (4) of ligand atoms in a

tetrahedral environment. The Co^2þion has a larger ꢀE than

the Fe^2þion. This is inconsistent with the known spec-

trochemical series for octahedral ligand ﬁelds; upon chang-

ing the center ions with ﬁxed ligand atoms, ꢀE is known to

400

200

0

1500

1000

500

0

(e)

CoCr₂O₄

(f)

CoCr₂S₄

increase in the following sequence: Mn^2þ, Ni^2þ, Co^2þ, Fe^2þ

,

1.5

2

2.5 1

1.5

Photon Energy (eV)

2

Photon Energy (eV)

V^2þ, Fe^3þ, Cr^3þ, V^3þ, Co^3þ, and Mn^4þions.¹²⁵⁾The spin–

orbit coupling constants ꢀ for the Fe^2þand Co^2þions are

rather large compared with those observed for the free A^2þ

ions.¹²⁷⁾This is also quite in contradiction to the established

understanding of the covalency eﬀect;¹²⁷⁾however, the

reason for this discrepancy is unclear at present. Meanwhile,

the A dependence of the magnitude of ꢀ coincides with that

of free ions.

Fig. 5. Optical conductivity (ꢆ) spectra for intra-atomic d–d transitions of

Fe^2þand Co^2þions in ACr₂X₄(A ¼ Fe and Co; X ¼ O and S). The solid

lines represent the results of ﬁtting with the multicomponent Lorentz

oscillator model; the dotted lines show the contributions of each

oscillator. We adopted a three-component model, taking account of level

splitting by spin–orbit coupling, for (a) the ⁵E ! T₂transition in

5

FeCr₂O₄, (b) that in FeCr₂S₄, (c) the ⁴A₂! T₂transition in CoCr₂O₄,

4

and (d) that in CoCr₂S₄. We add two other components—the intra-atomic

d–d transition of the Cr^3þion indicated by the open triangle and the Cr

charge transfer excitation giving rise to the slanted background—to the

The intra-atomic d–d transitions of A^2þions are also

clearly observed in the chalcogenides, while those associated

with Cr^3þions are barely discerned [Fig. 3(b)]. The spectra

are analyzed similarly to those of the oxides with few

exceptions. We presume S₁, S₂, and S₃as independent

three-component model for ﬁtting (e) the ⁴A₂! T₁transition in

4

CoCr₂O₄. Since ﬁne structures are unclear in (f) the ⁴A₂! T₁transition

4

in CoCr₂S₄, we employ a model with two oscillators, one of which

expresses the Cr charge transfer excitation.

034713-6

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

Table II. Crystal ﬁeld splitting ꢀE, Racah parameter B, and spin–orbit

coupling constant ꢀ of Fe^2þand Co^2þions. The data for free ions is cited

from the literature.^127ÞThe values for X ¼ O and S are deduced from the

analysis of the d–d transition spectra of ACr₂X₄(A ¼ Fe and Co; X ¼ O

and S) with the use of the ligand ﬁeld theory.

1

CoCr₂X₄

FeCr₂X₄

(a)

(b)

0.5

Free

O

S

ꢀE(Fe^2þ) (eV)

ꢀE(Co^2þ) (eV)

0

0.488

0.843

743

0.321

0.720

391

B(Co^2þ) (cm^ꢄ1

ꢀ(Fe^2þ⁵T₂) (cm^ꢄ1

)

971

410

533

0

1

,

)

1070

1992

2706

596

ꢀ(Co^2þ

,

⁴T₂) (cm^ꢄ1

⁴T₁) (cm^ꢄ1

)

1873

—

CoCr₂X₄

0.5

variables in the ﬁtting of the ⁵E ! T₂transition in FeCr₂S₄,

because the ﬁtting with S₁¼ 3S, S₂¼ 5S, and S₃¼ 7S has

resulted in a poor ﬁtting quality; this is probably related to

the energy-dependent mixing of Fe 3d orbitals with S 3p

orbitals. We adopt the single-component Lorentz oscillator

5

0

2

CoCr₂X₄

FeCr₂X₄

(c)

4

model for the ⁴A₂! T₁transition in CoCr₂S₄, because the

1

0

spin–orbit splitting is indiscernible. The ﬁtting results are

shown in Figs. 5(b), 5(d), and 5(f), and the deduced optical

parameters are listed in Table II.

Free

O

S

We discuss the covalency eﬀect by comparing the results

for the oxides with those for the chalcogenides in four

respects: (1) Oscillator strength. As can be clearly seen

from Fig. 5, the oscillator strength of the d–d transition for

chalcogenides is about 5 times larger than that for oxides;

this is caused by the spectral weight transfer in the optical

conductivity spectra for about 10 eV (Fig. 2). (2) Crystal

ﬁeld splitting ꢀE. For both A ¼ Fe and Co, ꢀE is larger

in the oxides than in the chalcogenides [Fig. 6(a)]. This

accords with the spectrochemical series for octahedral

ligand ﬁelds: The ꢀE for the ﬁxed center ion increases

X

Fig. 6. Ligand (X) dependence of (a) the crystal ﬁeld splitting ꢀE, (b) the

Racah parameter B, and (c) the spin–orbit coupling constant ꢀ deduced

from the analysis of the intra-atomic d–d transitions of the Fe^2þand

Co^2þions in ACr₂X₄(A ¼ Fe and Co; X ¼ O and S). ‘‘Free’’ in the

abscissa axis stands for the free ion state; their values are cited from the

literature.¹²⁷⁾

with the change in the ligand atoms in the order of I, Br, typical example, we present the optical conductivity spectra

Cl, S, F, O, N, and C.¹²⁵⁾ꢀE consists of the contribution of of CoCr₂S₄at various temperatures in Fig. 7(a). Upon

the ionic ligand ﬁeld ꢀE_ionand a correction term from the decreasing temperature, the two absorption peaks observed

covalency eﬀect ꢀE_cov(ꢀE ¼ ꢀE_ionþ ꢀE_cov).¹²⁷⁾Since are gradually sharpened without any abrupt change at T_c. At

ꢀE_ionis smaller and ꢀE_covis larger in the chalcogenides the lowest temperature measured, ﬁne structures, which are

4

with a more covalent nature than in the oxides with a more indiscernible at room temperature, appear in the A₂! T₂

ionic nature, we speculate that ꢀE_ionhas a major contribu- transition; these are most likely ascribed to the exchange

tion to ꢀE in the present compounds. (3) Racah parameter splitting brought about by the ferrimagnetic transition at T_c.

B. The B of the Co^2þion exhibits a considerable ligand In principle, it is possible to estimate the magnitude of the

dependence [Fig. 6(b)]; the B values of X ¼ O and S are 77 exchange splitting by ﬁtting the spectrum with the six-

and 40% of the free-ion value, respectively.¹²⁷⁾This is component Lorentz oscillator model. Keeping in mind that

qualitatively in accordance with the proposed nephelauxetic a ﬁtting with too many parameters reduces the reliability

series for octahedral ligand ﬁelds: The renormalization of B of the result, we analyze the low-temperature spectrum for

4

is enhanced by altering the ligand atoms in the order of F, O, the ⁴A₂! T₂transition in the same way as the room-

4

N, Cl, C, Br, S, and I.^125,128)The Wannier function is so temperature spectrum. The ⁴A₂! T₁spectrum, on the

expanded in the chalcogenides owing to the strong mixing of other hand, shows a marked red shift; these are also ﬁtted

Co 3d orbitals with S 3p orbitals that the local electron with the function used at room temperature. The thus-

correlation eﬀect is markedly weakened. (4) Spin–orbit obtained optical parameters are plotted against temperature

coupling constant ꢀ. Not only the two-electron operator B in Figs. 7(b)–7(f), where we introduce S_total¼ S₁þ S₂þ S₃

4

but also the one-electron operator ꢀ is highly renormalized in and ꢅ_av¼ ðꢅ₁þ ꢅ₂þ ꢅ₃Þ=3 for the ⁴A₂! T₂transition.

chalcogenides;¹²⁷⁾this is actually observed in both A ¼ Fe ꢀE increases with decreasing temperature, because a

and A ¼ Co systems [Fig. 6(c)].

decrease in lattice constants increases both ꢀE_ionand

ꢀE_cov. ꢅ_avis smaller at lower temperatures owing to a

reduction in the degree of thermal broadening (coupling to

the lattice phonon). The growth of admixture of Co 3d

3.3 Temperature dependence of optical conductivity

spectra of CoCr₂S₄

The intra-atomic d–d transition spectra are expected to orbitals and S 3p orbitals with decreasing temperature

show a pronounced temperature dependence. Here, as a increases S_total, and ꢀ and B are renormalized to smaller

034713-7

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

0.73

(b)

(c)

(d)

(e)

CoCr₂S₄

0.725

⁴A₂⁴T₁

(a)

0.72

250

⁴A₂⁴T₂

200

293 K

150

350

250 K

300

200 K

150 K

100 K

250

2000

1800

1600

1400

400

50 K

10 K

(f)

500

0

380

360

340

0.5

1

1.5

0

100 200 300

Temperature (K)

Photon Energy (eV)

Fig. 7. (a) Temperature dependence of optical conductivity (ꢆ) spectra in the energy range of 0.5–1.5 eV of CoCr₂S₄. The right panel shows the

temperature dependences of (b) the crystal ﬁeld splitting ꢀE, (c) the oscillator strength S_total¼ S₁þ S₂þ S₃, (d) the averaged damping constant

4

ꢅ_av¼ ðꢅ₁þ ꢅ₂þ ꢅ₃Þ=3, (e) the spin–orbit coupling constant ꢀ deduced from the ⁴A₂! T₂spectrum for the Co^2þion, and (f) the Racah parameter B,

4

which is proportional to the energy diﬀerence between the ⁴A₂! T₂and ⁴A₂! T₁transitions.

values. Although the increase in S_total, as well as the decrease

in B, is reminiscent of the metallic state at low temperatures,

the compound is actually an intrinsic Mott insulator.

1500

1000

500

0

CuCr₂Se₄

200

4000

100

3.4 Metal–insulator transition in A ¼ Cu system

The enhanced spectral weight in the low energy range and

the highly renormalized B in the chalcogenides, which are

revealed in §3.1–§3.3, are intuitively interpreted in a single

band Hubbard model:

0

100 200 300

Temperature (K)

2000

CuCr₂Se₄

X

CuCr₂O₄

0

H ¼ ꢄt

c^y_iꢆc_jꢆþ U

n_i"n_i#;

ð3Þ

i

i¼j

where c^y_iꢆ(c_iꢆ) is the creation (annihilation) operator of an

electron with spin ꢆ at the i-th site, t the transfer integral, U

the on-site Coulomb interaction, and n_iꢆ¼ c^y_iꢆc_iꢆ. Within

this model, the large spectral weight (oscillator strength) of

the optical conductivity in the low energy range corresponds

to a large t. The reduced B in the chalcogenides signiﬁes that

U, which is a linear combination of Racah parameters (A, B,

and C),¹²⁷⁾is also renormalized to a smaller value. There-

fore, the electron correlation parameter t=U is widely varied

by changing the ligand atom from O to S (Se); this variation

0

1

2

Photon Energy (eV)

3

Fig. 8. Optical conductivity (ꢆ) spectra of CuCr₂X₄(X ¼ O and Se) at

room temperature. Note that the ordinate axis for X ¼ O is diﬀerent from

that for X ¼ Se. The closed and open triangles stand for intra-atomic d–d

transitions associated with Cu^2þand Cr^3þions, respectively (see text for

details). The inset shows the temperature dependence of resistivity of

CuCr₂Se₄.

may possibly induce a metal–insulator transition. We here correlation eﬀect in the small t=U system. There are some

discuss the optical conductivity spectra of CuCr₂X₄(X ¼ O characteristic spectral structures below 3 eV. Following the

and S), a prototypical system showing such a phenomenon. discussion of ACr₂O₄(A ¼ Mn, Fe, and Co), we ascribe the

The optical conductivity spectrum of CuCr₂O₄(Fig. 8) absorptions at 0–1.5 eV (marked with the closed triangles)

exhibits an insulating feature; this originates from the strong and 1.5–2.5 eV (marked with the open triangles) to the

034713-8

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

intra-atomic d–d transitions of the Cu^2þions [²Eðe⁴t₂⁵Þ !

0.3

0.2

0.1

0

4

²T₂ðe³t₂⁶Þ] and Cr^3þions (⁴A_2g! T_2g), respectively. An

(a)

interesting aspect is that both absorptions are split into two

bands with a signiﬁcant energy diﬀerence. These are most

likely results of a reduction of the local symmetry from

cubic to tetragonal due to the J–T structural transition at

850 K;^23–25)in the low-temperature phase, the site symme-

H = 0.01 T

FeCr₂O₄

CuCr₂O₄

MnCr₂O₄

CoCr₂O₄

tries at the Cu^2þand Cr^3þsites are 4m2 and 2=m, respec-

ꢁ

tively.²⁵⁾In accordance with this, the c=a in the unit of

original cubic lattice constants is 0.913 at room temperature,

being considerably smaller than unity.²⁵⁾

The spectrum of CuCr₂Se₄(Fig. 8), on the other hand,

shows a metallic behavior in agreement with the dc

resistivity shown in the inset of Fig. 8. This is understood

by considering that the Mott gap in CuCr₂O₄with a feature

of the O 2p ! Cu 3d charge transfer excitation is collapsed

in CuCr₂Se₄owing to an increased t=U as well as to an

upward energy shift of the p level in the chalcogenides.

According to a previous soft x-ray MCD,⁸⁴⁾CuCr₂Se₄is

classiﬁed as a metal with a negative charge transfer gap

(p-type metal) in the Zaanen–Sawatzky–Allen scheme.¹³⁵⁾

In the light of this knowledge, the Drude response below

0.1 eV is mainly due to conducting holes in the Se 4p band.

The broad hump structure in the infrared region (marked

with a closed triangle) is likely related to the intra-atomic

0

50

100

150

T (K)

0.4

0.3

0.2

0.1

0

(b)

H = 0.01 T

CdCr₂Se₄

MnCr₂S₄

CuCr₂Se₄

CoCr₂S₄

2

d–d transitions of the Cu^2þions (²E ! T₂), as observed

FeCr₂S₄

in CuCr₂O₄, suggesting that the Cu valence is slightly

deviated from one; neutron diﬀraction analysis and MCD

experiments actually indicate the ﬁnite but small magnetic

moments at the Cu site.^81,84)The pronounced shoulder

structure at around 1.7 eV (marked with an open triangle)

closely resembles the intra-atomic d–d transition of the Cr^3þ

ZnCr₂Se₄

(×10)

0

100

200

300

T (K)

Fig. 9. Temperature (T) dependence of magnetization (M) under mag-

netic ﬁeld (H) of 0.01 T for ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd;

X ¼ O, S, and Se). The magnetic ﬁeld is applied along the [111] direction

except for CuCr₂Se₄with H k ½110ꢆ. The data was collected in a cooling

run. The closed triangles indicate ﬁrst-order phase transitions below

magnetic transition temperatures. Note that both ordinate and abscissa

scales for (a) the oxides diﬀer from those for (b) the chalcogenides. The

data of ZnCr₂Se₄is 10 times magniﬁed for clarity.

4

ions (⁴A_2g! T_2g) in CdCr₂S₄.⁵⁹⁾

4. Magnetic Properties

4.1 Temperature and magnetic ﬁeld dependences of

magnetization

Figure 9 shows the temperature dependence of magnet-

ization for ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd;

X ¼ O, S, and Se) measured in a cooling run. A magnetic

ﬁeld of 0.01 T is applied parallel to the [111] axis except for varies with nonmagnetic A^2þions from A ¼ Zn to A ¼ Cd.

CuCr₂Se₄under a magnetic ﬁeld along the [110] axis. All This suggests that the sign and magnitude of Cr–Cr magnetic

the oxides investigated undergo phase transition at T_ctoward interactions are quite sensitive to the A ion.

a magnetically ordered state with a ﬁnite spontaneous

Isothermal magnetization curves along the [111] direction

magnetization. In A ¼ Mn, Fe, and Co, it is followed by a at 2 K are shown in Fig. 10. The magnetization for all oxides

ﬁrst-order phase transition (marked with closed triangles) measured and MnCr₂S₄gradually develops with increasing

to the spin state with a long wave modulation vector magnetic ﬁeld. This, together with the smaller spontane-

(conical spin structure), which has been revealed by neutron ous magnetic moments than the value expected from the

diﬀraction analysis.^5–11)There is one ﬁrst-order transition ferrimagnetic arrangements between A^2þand Cr^3þspins,

at T⁰(< T_c) in A ¼ Mn and Fe, whereas there are two indicates the noncollinear spin arrangements due to the

in A ¼ Co; the second transition at T⁰⁰(< T⁰) has been competing magnetic interaction of A–Cr and Cr–Cr bonds.

identiﬁed as a lock-in transition.^9,11)These three transition At ﬁrst glance, FeCr₂S₄seems to be categorized into the

temperatures are insensitive to the magnitude of the applied same class as MnCr₂S₄. However, the magnetization reaches

ﬁeld. The chalcogenides with magnetic A^2þions (A ¼ Mn, a saturation value of 1.7 ꢇ_Bper formula unit at around 7 T

Fe, Co, and Cu) exhibit basically similar features. However, (data not shown), where the energy gain due to the Zeeman

the ﬁrst-order phase transition is discernible only for coupling overcomes the loss of magnetic anisotropy energy;

A ¼ Mn. We note that the decrease in the magnetization FeCr₂S₄is a ferrimagnet with antiparallel arrangements

of FeCr₂S₄below 60 K is not caused by a thermodynamic between Fe^2þand Cr^3þspins.⁶⁾The data of CoCr₂S₄show

phase transition; instead, it corresponds to the emergence of similar characteristics to FeCr₂S₄with a smaller saturation

magnetic anisotropy with the [111] direction as the hard axis magnetic ﬁeld ꢃ0:5 T. CuCr₂Se₄also shows a gradual

(see §4.2). The uniform magnetization of the chalcogenides saturation below 0.4 T; this is relevant not to the magnetic

034713-9

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

a ferromagnet or a ferrimagnet with high T_c, whereas a

compound with a large optical gap has a noncollinear spin

structure with a low T_c. This statement is based on the

following three considerations. (1) When we compare the

oxides with the chalcogenides for a ﬁxed A cation, we notice

that the chalcogenides have a higher T_cand larger mag-

netic moments (equivalently, smaller canting angles). It is

technically diﬃcult to estimate the optical gap of A ¼ Mn by

reﬂectivity measurement. As we discussed in §3, however,

the chalcogenides have larger spectral weights in the low-

energy optical conductivity spectra than the oxides. Thus,

MnCr₂O₄has a larger optical gap than MnCr₂S₄. The optical

gap of A ¼ Fe and Co is equal to ꢀE, which is larger in the

oxides than in the chalcogenides. It is clear that insulating

CuCr₂O₄has a larger optical gap than metallic CuCr₂Se₄.

(2) Magnetic transition temperature for the ﬁxed X anion

decreases in the order of A ¼ Cu, Co, Fe, and Mn; this

closely correlates with the increasing sequence of the lowest

d–d transition energy. The slight inconsistency between the

two series is explained convincingly as follows: A strong

hybridization between Co^2þ- and Cr^3þ-related bands with

essentially the same symmetries (⁴A₂and ⁴A_2g, respectively)

renders the T_cof the A ¼ Co system high; the Cu charge

transfer excitation also governs the optical gap of the A ¼

Cu system. (3) Concerning the systems with nonmagnetic

A^2þions, ZnCr₂Se₄is a helical magnet with T_c¼ 22 K,

whereas CdCr₂Se₄is a ferromagnet with T_c¼ 135 K.

Unfortunately, it is impossible to estimate the optical gap

of these compounds precisely from our reﬂectivity data.

Nevertheless, the optical absorption spectra in the literature

indicate that the optical gaps are 1.34 eV for ZnCr₂Se₄⁶²⁾and

1.32 eV for CdCr₂Se₄.⁵⁵⁾Moreover, we can refer to the

optical gap of the II–VI semiconductors ASe, because the

optical conductivity spectra in the low energy range for

ACr₂Se₄(A ¼ Zn and Cd) have a large contribution from the

Se 4p ! Zn 4s=4p and Cd 5s=5p transitions (see §3.1). The

energy gaps are 2.8 eV for ZnSe and 1.8 eV for CdSe.¹²¹⁾

From these results, we conclude that a ferromagnetic state is

1.5

T = 2 K, H || [111]

MnCr₂O₄

(a)

1

FeCr₂O₄

CuCr₂O₄

0.5

CoCr₂O₄

0

1

2

3

4

5

H (T)

T = 2 K, H || [111]

(b)

6

4

2

0

CuCr₂Se₄

CdCr₂Se₄

ZnCr₂Se₄

CoCr₂S₄

MnCr₂S₄

FeCr₂S₄

0

1

2

3

4

5

H (T)

Fig. 10. Magnetic ﬁeld (H) dependence of magnetization (M) along the

[111] direction at 2 K for ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd;

X ¼ O, S, and Se). The data were acquired with decreasing H. Note that

the ordinate scale for (a) the oxides is diﬀerent from that for (b) the

chalcogenide.

anisotropy, but rather to the demagnetizing eﬀect in a thin stabilized in a system with a small optical gap.

crystal. The magnetic structure of CuCr₂Se₄had been

simply assigned to the ferromagnetic alignment of Cr^3:5þ4.2 Magnetic anisotropy in chalcogenides

spins in Cu^þCr₂^3:5þSe₄^2ꢄ. However, neutron diﬀraction

Figure 11 shows torque curves at various temperatures for

analysis⁸¹⁾and MCD⁸⁴⁾experiment have revealed that the ACr₂S₄(A ¼ Mn, Fe, and Co) and CuCr₂Se₄. The external

actual valence state is Cu^ð1þpÞþCr₂^{ð7=2ꢄqÞþ}Se₄

magnetic ﬁeld of 7 T is rotated within the (110) plane. The

ð2þp=4ꢄq=2Þꢄ

with ﬁnite p and q, and that the induced spins at the Cu and magnetic torque L is a ﬁrst derivative of the magnetic

Se sites are antiparallel to the Cr spins. Polarized neutron anisotropy energy E_aniby the angle between the magnetic

diﬀraction analysis shows p ¼ 0:07 and q ¼ 0:14,⁸¹⁾while ﬁeld and crystal axes ꢈ, L ¼ ꢄ@E_ani=@ꢈ. The E_aniof a cubic

the MCD experiment shows q ¼ 0:43.⁸⁴⁾For arbitrary p and ferromagnet is expressed as a series of expansions in

q values, this ferrimagnetic alignment gives a saturated ascending powers of normalized magnetization as

magnetic moment of 5 ꢇ_Bper formula unit. This is con-

E_ani¼ K₁ðꢂ₁²ꢂ₂²þ ꢂ²₂ꢂ²₃þ ꢂ²₃ꢂ₁²Þ þ ꢇ ꢇ ꢇ ;

ð4Þ

sistent with the reported value of 5.1 ꢇ_B;⁷⁶⁾however, it

slightly deviates from our value of 5.4 ꢇ_B. The reason for where K₁is the magnetic anisotropy constant and ꢂ_ithe

this discrepancy is probably related to Se deﬁciency. The direction cosine of normalized magnetization along the i-th

magnetization of helical ZnCr₂Se₄is easily polarized by axis.¹³⁶⁾We ﬁt experimental torque curves with a leading

applying a magnetic ﬁeld, indicating that the system is term of eq. (4). Although 7 T is not enough to saturate the

very close to the ferromagnetic state, which is actually magnetization of MnCr₂S₄with the spin canted state, we

realized in CdCr₂Se₄.

analyze the data assuming that the spins are fully polarized.

The magnetic transition temperature and magnetic struc- Fitting quality is somewhat low for FeCr₂S₄, indicating a

ture summarized in Table I are well recapitulated in terms of sizeable contribution from higher-order terms. The obtained

the optical gap investigated by optical reﬂectivity spec- K₁is consistent with reported values,^{76,79,80,137–140)}and is

troscopy: A compound with a small optical gap is typically plotted as a function of temperature scaled by T_cin Fig. 12.

034713-10

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

signs of K₁for these three compounds. Since L belongs to

the T_1grepresentation, only the excited state containing T_1g

in the product representation with the ground state contrib-

utes to magnetic anisotropy. Moreover, j0i and jni should be

indexed by the same spin quantum number, because L is a

spin-independent operator. The absence of such an excited

state in A ¼ Mn leads to a relatively small K₁. In FeCr₂S₄

and CoCr₂S₄, the process that most strongly contributes to

(110) plane,

7 T

H =

[10³]

6

4

2

0

5 K

(a)

10

15

2025

30

35

40 K

5

magnetic anisotropy is the E ! T₂transition (⁵E ꢅ T₂¼

MnCr₂S₄

5

⁵T₁þ T₂) and ⁴A₂! T₂transition (⁴A₂ꢅ T₂¼ ⁴T₁),

respectively; both transitions are observed in the reﬂectivity

measurements. The transition energies ꢀE₁for these

excitations are identical with the crystal ﬁeld splitting

energy ꢀE, and are 0.488 eV for FeCr₂S₄and 0.843 eV for

CoCr₂S₄. This variation in ꢀE, together with the expres-

sion of K₁[eq. (5)], well explains the stronger magnetic

anisotropy in FeCr₂S₄than in CoCr₂S₄. The large negative

magnetic anisotropy constant of CuCr₂Se₄, on the other

hand, can hardly be interpreted in terms of the second-order

perturbation of spin–orbit coupling in A^2þions. Instead, it is

likely relevant to unquenched orbital magnetic moments of

5

4

2

0

[10 ]

5 K

20

30

(b)

50

40

60

70

80

90 K

FeCr₂S₄

[10⁴]

4

2

0

5 K

30

(c)

60

90

120

150

180

210 K

ð2þp=4ꢄq=2Þꢄ

Cu^2þand Cr^4þions in Cu^ð1þpÞþCr₂^{ð7=2ꢄqÞþ}Se₄

.

Neutron diﬀraction analysis demonstrated that the Cr

valence is 3.36 (q ¼ 0:14) and that the Cu valence is 1.07

(p ¼ 0:07),⁸¹⁾suggesting that Cr atoms are responsible for

the observed magnetic anisotropy. However, the application

of the sum rule to the MCD spectra revealed very small

orbital magnetic moments at Cr sites.⁸⁴⁾Therefore, which

of two atoms gives rise to the large magnetic anisotropy

observed here remains to be solved.

CoCr₂S₄

4

6

4

2

0

[10 ]

5 K

(d)

40

80

120

160

200

240 K

We discuss the temperature dependence of K₁. According

to phenomenological arguments, K₁changes its magnitude

in such a way that K₁/ m¹⁰, where m is the magnetization

normalized by the saturation magnetization.^136,143)Our

experimental data, however, show a large variation rang-

ing from a steep increase with decreasing temperature in

FeCr₂S₄to a gradual one in CoCr₂S₄. This deviation from

the m¹⁰law is widely observed for various reasons.¹³⁶⁾We

notice that the K₁of FeCr₂S₄slightly decreases near 5 K

(Fig. 12); this marks the Jahn–Teller structural transforma-

tion associated with Fe^2þions.^104–106)

CuCr₂Se₄

240

Position (deg)

0

120

360

Fig. 11. Magnetic torque curves within (110) plane at various temper-

atures for (a) MnCr₂S₄, (b) FeCr₂S₄, (c) CoCr₂S₄, and (d) CuCr₂Se₄. The

experiments are performed under a magnetic ﬁeld (H) of 7 T. Note that

the scales of each ordinate axis are not identical.

We ﬁrst focus on the sign and magnitude of K₁in the

ground state. Although K₁is positive (with the [100]

5. Magnetooptical Properties

direction being the easy axis) for ACr₂S₄(A ¼ Mn, Fe, and 5.1 Overall feature of magnetooptical spectra of

Co), it is negative for CuCr₂Se₄with a considerably large

magnitude (with the [111] direction being the easy axis).

This implies that the mechanism of magnetic anisotropy is rotation (ꢈ) and ellipticity (ꢉ) in the photon energy range

chalcogenides

Figure 13 shows the spectra of magnetooptical Kerr

distinguishable between the two classes. Considering the of 0.2–4.5 eV for ACr₂S₄(A ¼ Mn, Fe, and Co) and

138,141)

isotropic nature of Cr^3þions in CdCr₂Se₄

as well as ACr₂Se₄(A ¼ Cu, Zn, and Cd). Our spectrum of CoCr₂S₄

the quenched orbital moments of Mn^2þ, Fe^2þ, and Co^2þrecorded on the as-grown (111) surface at ꢃ0:2 T basically

ions,¹²⁷⁾we attribute the magnetic anisotropy of ACr₂S₄agrees with the reported spectrum for the pressed pel-

(A ¼ Mn, Fe, and Co) to the second-order perturbation of the let.^98–101)Data for ACr₂Se₄(A ¼ Cu and Cd) agree with

spin–orbit coupling in A^2þions, which is expressed as

those in the earlier studies.^{40,61,63,64,102,103)}The CuCr₂Se₄

spectrum is also consistent with those calculated from ﬁrst

principles.⁸⁶⁾

X

h0jLjni ꢇ hnjLj0i

ꢀE_n

K₁/ ꢄ²

;

ð5Þ

n

The magnetooptical Kerr eﬀect spectra of ACr₂S₄

where ꢄ is a constant proportional to ꢀ, L represents the (A ¼ Mn, Fe, and Co) and CuCr₂Se₄above 2.0 eV resemble

orbital angular momentum operator, and ꢀE_nis the energy each other, because the Cr charge transfer excitation is

diﬀerence between the ground state j0i and the n-th excited responsible for absorptions in this energy range. Below

state jni.¹⁴²⁾It is noteworthy that the sign of K₁does not 2.0 eV, each system shows characteristic spectral structures

depend on that of ꢄ. This well accounts for the identical depending on A atoms. The ﬁne structure at around 1.8 eV in

034713-11

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

magnetooptical features of MnCr₂S₄and FeCr₂S₄in a

separate paper.⁹⁶⁾In the present paper, we discuss those of

CoCr₂S₄in the following subsection. The CuCr₂Se₄spec-

trum has a pronounced structure at around 1.0 eV; this is

identiﬁed as the plasma resonance in a metal.^102,103,144)The

overlap of the Se 4p ! Zn 4s=4p and Cd 5s=5p transitions

with the Cr charge transfer transition (§3.1) makes the

spectra for ACr₂Se₄(A ¼ Zn and Cd) extremely complicat-

ed. It should be noted that the CdCr₂Se₄spectrum with a

slight upward energy shift (ꢃ0:4 eV) and rounding bear

close resemblance to the ZnCr₂Se₄spectrum. This energy

shift corresponds to the energy diﬀerence between the

Se 4p ! Zn 4s=4p and Cd 5s=5p transitions (see the dis-

cussion in §4.1). The blurred spectral structures of ZnCr₂Se₄

may originate from the unsaturation of the magnetic mo-

ments.

2

FeCr₂S₄(×1/3)

MnCr₂S₄

1

0

CoCr₂S₄

–1

–2

CuCr₂Se₄

0

0.5

T/T_c

1

Fig. 12. Magnetic anisotropy constant K₁as a function of the temperature

(T) normalized by the magnetic transition temperature (T_c) for ACr₂S₄

(A ¼ Mn, Fe, and Co) and CuCr₂Se₄. Note that the K₁values for FeCr₂S₄

are multiplied by a factor of 1/3 for clarity.

5.2 Magnetooptical spectra of CoCr₂S₄

Figure 14 shows the magnetooptical spectra of CoCr₂S₄

at various temperatures. The peak value of ꢈ reaches 4.1^ꢁ

at the d–d transition energy, being much larger than the

MnCr₂S₄is due to resonance with the spin-forbidden d–d conventional values of magnetooptical materials (typically

4

transition of Mn^2þions ⁶A₁ðe²t₂³Þ ! T₁ðe³t₂²Þ, which is 0.5^ꢁ).^145,146)According to the same arguments presented in

indiscernible in the optical reﬂectivity spectrum.⁹⁶⁾Con- the preceding paper for FeCr₂S₄,⁹⁶⁾we conclude that the

spicuous structures below 2.0 eV in FeCr₂S₄and CoCr₂S₄origin of the gigantic signal is the large oscillator strength

are attributable to the intra-atomic d–d transition of A^2þcaused by the broken inversion symmetry at the tetrahedral

5

4

ions, the ⁵E ! T₂transition in FeCr₂S₄, and the ⁴A₂! T₂site and the large covalency eﬀect in the chalcogenides.

4

and A₂! T₁transitions in CoCr₂S₄. Since these transi-

In order to gain microscopic insight into the transition

tions are spin-allowed, the resonance feature is magniﬁed mechanism, we deduce oﬀ-diagonal components of the

by one order of magnitude in comparison with the 1.8 eV dielectric constant "_xyby substituting "_xxacquired by the

structure of MnCr₂S₄. We have already described the reﬂectivity measurement to the expression

MnCr₂S₄

FeCr₂S₄

CoCr₂S₄

4

2

4

2

0

–0.1

–0.2

0

–2

10 K

–0.3

0.2

–4

4

–4

4

2

0

2

0

–2

–4

0

–2

–4

0

Fig. 13. Magnetooptical Kerr rotation (ꢈ) and ellip-

ticity (ꢉ) spectra of ACr₂S₄(A ¼ Mn, Fe, and Co)

and ACr₂Se₄(A ¼ Cu, Zn, and Cd). The data at

10 K are shown except for CuCr₂Se₄, for which

experiments are performed at room temperature

(RT). Note that the ordinate scales of the respective

panels are not identical.

–0.2

0

2

4

2

4

2

4

Energy (eV)

CuCr₂Se₄

ZnCr₂Se₄

CdCr₂Se₄

0.4

0

0.5

0

–0.5

–0.05

0.05

RT

10 K

–0.4

0.4

–1

0.5

0

–0.5

–1

0

–0.4

–0.05

0

2

4

0

2

4

0

2

3

Energy (eV)

034713-12

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

CoCr₂S₄

0.2

0

2

0

4

2

10 K

50 K

100 K

150 K

200 K

215 K

220 K

(a)

200 K

150 K

0

–2

–4

4

100 K

50 K

10 K

2

0

–2

–4

–6

–4

–0.2

0.2

0.5

1

0.5

1

1.5

2

3

4

Photon Energy (eV)

(b)

10 K

50 K

4

2

(b)

100 K

150 K

200 K

215 K

220 K

0

4

0

2

0

200 K

150 K

100 K

50 K

–2

–4

10 K

0.5

1

–4

0.5

1

1.5

–0.2

0.5

1

1.5

2

3

4

Photon Energy (eV)

Fig. 15. Temperature dependences of spectra of (a) the real and (b)

imaginary part of "_xyfor the intra-atomic d–d transition of Co^2þions in

Fig. 14. Temperature dependences of spectra of (a) the Kerr rotation (ꢈ)

and (b) ellipticity (ꢉ) in energy range of 0.5–4.5 eV for CoCr₂S₄. The

abscissa scales are magniﬁed for the intra-atomic d–d transition of Co^2þ

4

CoCr₂S₄. The insets show the ﬁtting results for the ⁴A₂! T₂spectrum

at 10 K (see text for details).

4

ions, ⁴A₂! T₂and ⁴A₂! T₁. Note that the ordinate scales for 0.5–

1.5 eV and 1.5–4.5 eV are diﬀerent.

4

the A₂! T₂transition is somewhat complicated. Hence,

4

referring to the results for the ⁴A₂! T₁transition, we

"

xy

ꢈ þ iꢉ ¼

:

ð6Þ

pﬃﬃﬃﬃﬃﬃ

adopt the three-component model with !₁¼ ꢀE ꢄ ð1=4Þꢀ,

ð1 ꢄ "_xxÞ "_xx

!₂¼ ꢀE þ ð1=6Þꢀ, !₃¼ ꢀE þ ð5=12Þꢀ, S^þ₁¼ S, S₁^ꢄ¼ 10S,

þ

The thus-obtained "_xyspectra for the A₂! T₂and A₂! S^þ₂¼ 4S, S₂^ꢄ¼ 0, S₃¼ 5S, and S₃^ꢄ¼ 0. The ﬁtting result

⁴T₁transitions are plotted in Fig. 15. Both transitions exhibit for T ¼ 10 K is shown in the insets of Fig. 15; this yields

the diamagnetic-type spectra; however, the damping eﬀect ꢀE ¼ 0:719 eV and ꢀ ¼ 1480 cm^ꢄ1, which are consistent

due to coupling with the lattice vibration is too energy- with the values deduced from the optical conductivity

4

dependent that the spectra have an asymmetric shape with a spectra. A similar analysis of the A₂! T₁transition is

broad structure on the high-energy side. We notice that the precluded by the large damping eﬀect.

spectra for the two transitions have opposite signs: The real

In order to analyze the temperature dependence, we plot

part of " for the ⁴A₂! T₂(⁴A₂! T₁) transition has the temperature variation of the maximum Kerr rotation ꢈ in

negative (positive) values on the low-energy side and the measured energy range (ꢈ_max) as well as the magnet-

positive (negative) values on the high-energy side. This ization along the [111] axis at 0.2 T in Fig. 16. ꢈ_maxis

4

diﬀerence in sign is ascribable to the distinct irreducible located in the ⁴A₂! T₂transition (ꢃ0:7 eV), thereby

4

representation of excited states; the matrix elements of the reﬂecting the sublattice magnetization of Co^2þspins.⁹⁷⁾The

4

equivalent operator T of the T₂and T₁are opposite in ꢈ_maxand magnetization exhibit a good linear relation; hence,

sign and identical in absolute value.^127,147)We ﬁt the "_xymagnetooptical Kerr eﬀect spectroscopy is a useful tool for

4

spectrum for the A₂! T₂transition with the multicom- monitoring the total magnetization in the present compound.

ponent Lorentz oscillator model

It should be noted that this linear relation is maintained by

the proportionality between the sublattice magnetization of

Co^2þspins and the total magnetization. Generally, magneto-

optical Kerr eﬀect signals do not necessarily respond to the

S^þ_jꢄ S^ꢄ_j

X

2ꢃ

"

¼

!

j

;

ð7Þ

xy

2

!²_jꢄ ð! þ iꢅ_jÞ

i

j

which is deduced from the Kubo formula with the total magnetization; for example, ꢈ_maxdoes not scale with

approximation of ! ꢃ !_j.¹⁴⁸⁾Here, S_j^þand S^ꢄ_jare the the total magnetization in FeCr₂S₄, where the spectrum in

oscillator strengths of the j-th oscillator for the right- and the speciﬁc intra-atomic d–d transition energy reﬂects the

left-handed circular-polarized light, respectively. As dis- sublattice magnetization of Fe^2þspins⁹⁷⁾and/or is seriously

cussed in §3.2, the calculation of the oscillator strength for aﬀected by the electronic state of the relevant excitations.⁹⁶⁾

034713-13

J. Phys. Soc. Jpn., Vol. 77, No. 3

K. OHGUSHI et al.

tetrahedral site; this oﬀers a guiding principle in our search

for ideal magnetooptical materials.

CoCr₂S₄

2

1

0

4

2

H || [111]

H = 0.2 T

Acknowledgment

This work was partly supported by the Joint Studies

Program of the Institute for Molecular Science. We wish

to acknowledge enlightening discussions with T. Arima, S.

´

Horiuchi, I. Kezsmarki, T. Katsufuji, R. Kumai, J. Matsuno,

S. Nagata, T. Okuda, Y. Taguchi, and Y. Tomioka. We are

also grateful to M. Matsubara for his assistance in measuring

magnetooptical spectra.

0

100

200

300

T (K)

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Fig. 16. Temperature (T) dependences of magnetization (M) and max-

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)

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along the [111] axis. ꢈ_maxis located at around 0.7 eV in CoCr₂S₄; this

4

corresponds to the ⁴A₂! T₂transition of Co^2þions. The dotted line for

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6. Conclusion

We have carried out a systematic study of the magnetic,

optical, and magnetooptical properties of single-crystalline

ACr₂X₄(A ¼ Mn, Fe, Co, Cu, Zn, and Cd; X ¼ O, S, and

Se). We clariﬁed that the optical gaps of these compounds

are governed by A-related transitions, not by Cr-related ones.

We elucidated the covalency eﬀect by analyzing the optical

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atoms (X ¼ O and Se). The relationship between magnetism

and optical gap was revealed. A system with a large optical

gap typically undergoes a magnetic transition into noncol-

linear spin arrangements at a relatively low temperature,

whereas a system with a small optical gap is a collinear

ferromagnet or ferrimagnet with a rather high transition

temperature. This raises the possibility of acquiring high-T_c

materials by suitable carrier doping. One naturally realized

example is CuCr₂Se₄with a remarkably high transition

temperature (430 K). We successfully accounted for the

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4

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4

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4

by ⁴A₂ð FÞ, ⁴T₂ð FÞ, ⁴T₁ð FÞ, and ⁴T₁ð PÞ. In most of the earlier

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4

and ⁴A₂ð FÞ ! T₁ð PÞ transitions; this assignment contradicts

147) The sign of the "_xyspectrum for the ⁵E ! T₂transition in FeCr₂S₄

4

5

4

ours. According to the earlier interpretation, there is another lower

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4

034713-16

Article Doi

Magnetic, optical, and magnetooptical properties of spinel-type ACr 2X4 (A = Mn, Fe, Co, Cu, Zn, Cd; X = O, S, Se)

DOI: 10.1143/JPSJ.77.034713

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