theory of multiple spin density wave and lattice distortion in fcc antiferromagnets

6
Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets Yasuyuki MATSUURA and Takeo JO y Department of Quantum Matter, ADSM, Hiroshima University, Higashihiroshima, Hiroshima 739-8530 (Received August 19, 2009; accepted October 19, 2009; published December 10, 2009) The phase diagram of the Mn 1x A x (A ¼ Ni, Ga, Rh, and Au) alloys, which are known as the first-kind antiferromagnets, is explained theoretically. Our Hamiltonian is composed of the polynomial of the variables describing multiple spin density wave (MSDW) states, the coupling between the variables and the symmetry strain and the elastic energy. The polynomial is derived from symmetry consideration. By calculating the partition function and the free energy, we show that the phase diagram is reproduced and elucidate the structure of the MSDW at each phase and the condition of the appearance of the orthorhombic phase. KEYWORDS: multiple spin density wave, first-kind antiferromagnet, magnetoelastic coupling, symmetry strain DOI: 10.1143/JPSJ.78.124709 1. Introduction A lot of crystals undergo lattice distortions from a high symmetry phase to a low symmetry one with varying the temperatures. Some mechanisms of the phenomenon have been proposed. For example, the cooperative Jahn–Teller (JT) effect 1) and the magnetoelastic coupling 2,3) are typical examples. The microscopic origin of the phenomenon is the coupling between the degree of freedom of the electronic system and the lattice system. There have been attempted several theoretical explanations of the phenomenon. The optimization of the lattice structure by the first-principles calculations is the typical approach. 4) There are, however, limited numbers of microscopic theories which explain the phase diagram in more than one external parameter-space such as temperature and concentration. Kataoka and Kanamori 5) proposed a microscopic theory of the cooperative JT effect to explain the phase diagram of Cu 1x Ni x Cr 2 O 4 in the temperature–concentration of Ni (T x) plane, which includes the cubic phase, the tetragonal ones with c=a < 1 and c=a > 1 and the orthorhombic one. 6) They assumed the Hamiltonian composed of the symmetrized variables in terms of the angular momentum of magnitude 1, the elastic energy described by the bulk symmetry strain and the coupling between the variables and the bulk symmetry strain. By calculating the trace with respect to the degenerate electronic states, they obtained the partition function and the free energy, and drew the phase diagram by minimizing the free energy with respect to the strain. The purpose of the present work is to present a micro- scopic theory of lattice distortions for the purpose of explaining the phase diagram observed in the antiferromag- netic region of some fcc alloys. All the nearest neighboring magnetic moment pairs in fcc antiferromagnets cannot be antiparallel. As a result, antiferromagnetic interaction causes lattice distortion. The fcc alloys Mn 1x A x (A ¼ Ni, Ga, Rh, and Au) 7–10) show the so-called first-kind antiferromagnetic structure below the Ne ´el temperatures (T N ). In the anti- ferromagnetic region of the alloys, the phase diagram of the lattice distortion including the cubic phase, the tetragonal ones with c=a < 1 and c=a > 1 and the orthorhombic one in the T x plane is reported; schematic phase diagram is shown in Fig. 1. The Mn 1x Ir x alloys have a similar phase diagram, 11) but the tetragonal (c=a < 1) phase is not confirmed because the -phase is unstable in the region of the low concentration of Ir. We discuss the magnetoelastic coupling, which is a similar level to the above theory of JT effect proposed by Kataoka and Kanamori. In the first-kind antiferromagnets, there are three wave vectors to describe the modulation of the atomic magnetic moments Q x ¼ð2%=aÞð1; 0; 0Þ, Q y ¼ð2%=aÞð0; 1; 0Þ, and Q z ¼ð2%=aÞð0; 0; 1Þ with the lattice constant a. The general magnetic structure is described by the superposition of the modulations with Q x , Q y , and Q z . This state is called a multiple spin density wave (MSDW) state. In the Single-Q state described by Q z , the magnetic moments align parallel in the xy plane and antiparallel along the z axis. If there exists the antiferromagnetic interaction between the nearest neighboring antiparallel magnetic moment pairs, it is expected that the lattice constant along the z axis decreases and the lattice constants along the x and y axes increase, i.e., the tetragonal distortion with c=a < 1 in the Single-Q state. In fact, the first-principles calculations for Mn, 4,12) which optimize the lattice dis- tortion, give the tetragonal (c=a < 1) lattice structure in the Single-Q state. Fig. 1. The schematic phase diagram of Mn 1x A x (A ¼ Ni, Ga, Rh, and Au) alloys. 7–10) E-mail: [email protected] y E-mail: [email protected] Journal of the Physical Society of Japan Vol. 78, No. 12, December, 2009, 124709 #2009 The Physical Society of Japan 124709-1

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Page 1: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

Theory of Multiple Spin Density Wave and Lattice Distortion

in fcc Antiferromagnets

Yasuyuki MATSUURA� and Takeo JO

y

Department of Quantum Matter, ADSM, Hiroshima University,

Higashihiroshima, Hiroshima 739-8530

(Received August 19, 2009; accepted October 19, 2009; published December 10, 2009)

The phase diagram of the Mn1�xAx (A ¼ Ni, Ga, Rh, and Au) alloys, which are known as the first-kindantiferromagnets, is explained theoretically. Our Hamiltonian is composed of the polynomial of thevariables describing multiple spin density wave (MSDW) states, the coupling between the variables andthe symmetry strain and the elastic energy. The polynomial is derived from symmetry consideration. Bycalculating the partition function and the free energy, we show that the phase diagram is reproduced andelucidate the structure of the MSDW at each phase and the condition of the appearance of theorthorhombic phase.

KEYWORDS: multiple spin density wave, first-kind antiferromagnet, magnetoelastic coupling, symmetry strainDOI: 10.1143/JPSJ.78.124709

1. Introduction

A lot of crystals undergo lattice distortions from a highsymmetry phase to a low symmetry one with varying thetemperatures. Some mechanisms of the phenomenon havebeen proposed. For example, the cooperative Jahn–Teller(JT) effect1) and the magnetoelastic coupling2,3) are typicalexamples. The microscopic origin of the phenomenon is thecoupling between the degree of freedom of the electronicsystem and the lattice system. There have been attemptedseveral theoretical explanations of the phenomenon. Theoptimization of the lattice structure by the first-principlescalculations is the typical approach.4) There are, however,limited numbers of microscopic theories which explain thephase diagram in more than one external parameter-spacesuch as temperature and concentration.

Kataoka and Kanamori5) proposed a microscopic theory ofthe cooperative JT effect to explain the phase diagram ofCu1�xNixCr2O4 in the temperature–concentration of Ni (T–x)plane, which includes the cubic phase, the tetragonal oneswith c=a < 1 and c=a > 1 and the orthorhombic one.6) Theyassumed the Hamiltonian composed of the symmetrizedvariables in terms of the angular momentum of magnitude 1,the elastic energy described by the bulk symmetry strain andthe coupling between the variables and the bulk symmetrystrain. By calculating the trace with respect to the degenerateelectronic states, they obtained the partition function and thefree energy, and drew the phase diagram by minimizing thefree energy with respect to the strain.

The purpose of the present work is to present a micro-scopic theory of lattice distortions for the purpose ofexplaining the phase diagram observed in the antiferromag-netic region of some fcc alloys. All the nearest neighboringmagnetic moment pairs in fcc antiferromagnets cannot beantiparallel. As a result, antiferromagnetic interaction causeslattice distortion. The fcc alloys Mn1�xAx (A ¼ Ni, Ga, Rh,and Au)7–10) show the so-called first-kind antiferromagneticstructure below the Neel temperatures (TN). In the anti-ferromagnetic region of the alloys, the phase diagram of the

lattice distortion including the cubic phase, the tetragonalones with c=a < 1 and c=a > 1 and the orthorhombic onein the T–x plane is reported; schematic phase diagram isshown in Fig. 1. The Mn1�xIrx alloys have a similar phasediagram,11) but the tetragonal (c=a < 1) phase is notconfirmed because the �-phase is unstable in the region ofthe low concentration of Ir. We discuss the magnetoelasticcoupling, which is a similar level to the above theory ofJT effect proposed by Kataoka and Kanamori. In thefirst-kind antiferromagnets, there are three wave vectorsto describe the modulation of the atomic magneticmoments Qx ¼ ð2�=aÞð1; 0; 0Þ, Qy ¼ ð2�=aÞð0; 1; 0Þ, andQz ¼ ð2�=aÞð0; 0; 1Þ with the lattice constant a. The generalmagnetic structure is described by the superposition ofthe modulations with Qx, Qy, and Qz. This state iscalled a multiple spin density wave (MSDW) state. Inthe Single-Q state described by Qz, the magnetic momentsalign parallel in the x–y plane and antiparallel alongthe z axis. If there exists the antiferromagnetic interactionbetween the nearest neighboring antiparallel magneticmoment pairs, it is expected that the lattice constantalong the z axis decreases and the lattice constants alongthe x and y axes increase, i.e., the tetragonal distortion withc=a < 1 in the Single-Q state. In fact, the first-principlescalculations for �Mn,4,12) which optimize the lattice dis-tortion, give the tetragonal (c=a < 1) lattice structure in theSingle-Q state.

Fig. 1. The schematic phase diagram of Mn1�xAx (A ¼ Ni, Ga, Rh, and

Au) alloys.7–10)

�E-mail: [email protected]: [email protected]

Journal of the Physical Society of Japan

Vol. 78, No. 12, December, 2009, 124709

#2009 The Physical Society of Japan

124709-1

Page 2: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

Jo and Hirai discussed theoretically the MSDW states andthe phase diagrams of fcc Mn alloys on the basis of Landau’sphenomenological theory.13) Fishman et al. gave a phenom-enological discussion on the phase diagram and MSDW.14) Inthe present work we introduce ‘‘microscopic’’ variables todescribe MSDW states. Our Hamiltonian is composed ofthe polynomial of the variables, the coupling between thevariables and the lattice strain and the elastic energy, all ofwhich are derived from symmetry consideration of oursystem. It contains some parameters as the coefficient of eachterm, which is, for example, to be determined by electronicstructure calculations. We calculate the partition function byintegrating with respect to the variables which describeMSDW states, and obtain the Helmholtz free energy as afunction of the lattice strain. The free energy gives the moststable lattice structure at each point in the T–x plane. InMn1�xAx (A ¼ Ni, Ga, Rh, and Au) alloys, the magneticmoment arises from 3d itinerant electrons, where theelectronic structure is to be taken into account explicitly.The explanation of the phase diagram (see Fig. 1) based onthe electronic state seems to be beyond the tractable limit atpresent. We expect our approach gives some insights intomore realistic ones based on the electronic structure. If weconfine ourselves to the absolute zero temperature, thefirst-principles Korringa–Kohn–Rostoker coherent potentialapproximation (KKR-CPA) calculations15) for example maygive the observed concentration dependence of MSDW statesand lattice distortion. Even if the calculations explain theexperimental results, a theory giving the relation between theMSDW states and lattice distortion at finite temperatures isstill lacking. If the concentration dependence of the param-eters of our model is estimated by the first-principles KKR-CPA calculations, the present work is expected to contributeto the understanding of the magnetism and the latticedistortion of Mn alloys. Some Mn alloys are used as anexchange biasing film in giant magnetoresistance (GMR)devices, and the first-principles calculations have beenattempted to investigate the magnetic structure of them.11,16)

We expect the present approach gives some insights into themagnetic properties of fcc first-kind antiferromagnets at finitetemperatures.

2. Hamiltonian

In the MSDW state where the component of Qi is Ai

(i ¼ x; y; z), the spin at the i-th lattice point Ri withmagnitude S is represented as

Si ¼ SðAxeiQx�Riiþ Aye

iQy�Ri jþ AzeiQz�RikÞ; ð1Þ

where i, j, and k are the unit vectors toward the x, y, andz axes, respectively. If we assume Ax ¼ sin � cos ’, Ay ¼sin � sin ’ and Az ¼ cos � by using the polar and azimuthalangles, the MSDW state is shown in Fig. 2. In the absence ofthe spin–orbit interaction, the spin space and the real spaceare independent with each other, and we can assume thelongitudinal modulation without the loss of generality. Wealso note that the present discussion also applies to MSDWstates where the magnetic moments shown in Fig. 2 arerotated uniformly with keeping their relative angles. Weimagine the Hamiltonian of the spin system described interms of Si’s, Helectron. Then the lowest order part is theclassical Heisenberg interaction and the next order part is the

so-called four-spin interaction and so on. We substitute themagnetic structure eq. (1) into Helectron. Then Helectron is bysymmetry consideration expanded as

Helectron ¼ �ðA2x þ A2

y þ A2z Þ þ �ðA

4x þ A4

y þ A4z Þ

þ �ðA6x þ A6

y þ A6z Þ þ � � � ð2Þ

with the coefficients �, �, and �, where we used therelation A2

x þ A2y þ A2

z ¼ 1. The first term of the right-hand side of eq. (2) corresponding to the classicalHeisenberg interaction, is found to be constant, that is,MSDW states are degenerate under the interaction and thedegeneracy is removed by the four-spin and higher orderinteractions.

We define X2 and X3 by

X2 �1ffiffiffi2p ðA2

x � A2yÞ; ð3Þ

X3 �1ffiffiffi6p ð2A2

z � A2x � A2

yÞ; ð4Þ

to describe the MSDW states in the first-kind fcc antiferro-magnets. Since Ax, Ay, and Az satisfy the relation A2

x þ A2y þ

A2z ¼ 1, X2, and X3 take the value inside or on the regular

triangle shown in Fig. 3. In addition to X2 and X3, we definethe symmetry strains "2 and "3 to describe the latticedistortion as follows:

"2 �1ffiffiffi2p ðexx � eyyÞ; ð5Þ

"3 �1ffiffiffi6p ð2ezz � exx � eyyÞ; ð6Þ

where exx, eyy, and ezz represent the strain along x, y, and z

axes, respectively.By symmetry consideration, we tentatively adopt the

following Hamiltonian

H ¼ Helectron þHcoupling þHelastic ð7Þ

with

Helectron ¼ AðX22 þ X2

3Þ þ BðX33 � 3X3X

22Þ

þ CðX33 � 3X3X

22Þ

2; ð8ÞHcoupling ¼ gð"2X2 þ "3X3Þ; ð9Þ

Helastic ¼�

2ð"22 þ "

23Þ; ð10Þ

Fig. 2. The multiple spin density wave state in fcc first-kind antiferro-

magnets. The directions of atomic moments are specified by (�; ’),

(�; �þ ’), (�� �; �’), and (�� �; �� ’) at atomic sites 1, 2, 3, and 4

respectively.

J. Phys. Soc. Jpn., Vol. 78, No. 12 Y. MATSUURA and T. JO

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Page 3: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

where Helectron, Hcoupling, and Helastic correspond to MSDWstates, the coupling between MSDW states and latticedistortion with the positive coupling constant g and theelastic energy with the elastic constant �, respectively. Thefirst and the second terms of eq. (8) correspond to the secondand the third terms of eq. (2), respectively, apart from theconstant term. In a later section, we discuss the origins of thelast term of eq. (8).

Generally, the leading magnetoelastic coupling term isproportional to the square of magnetization multiplied by thestrain. In eq. (9), X2 and X3 are the dimensionless variables.Then the coupling parameter g must be proportional to thesquare of the sublattice magnetization S.

3. Ground State and Scenario in Mn Alloys

In the ground state, by using the conditions @H=@"2 ¼ 0

and @H=@"3 ¼ 0, which minimize the Hamiltonian H withrespect to "2 and "3, we obtain

"2 ¼ �g

�X2; ð11Þ

"3 ¼ �g

�X3; ð12Þ

and the Hamiltonian is expressed as a function of thevariables X2 and X3

H ¼ A0ðX22 þ X2

3Þ þ BðX33 � 3X3X

22Þ

þ CðX33 � 3X3X

22Þ

2; ð13Þwith A0 ¼ A� g2=ð2�Þ. Equations (11) and (12) show thatthe Triple-Q state with A2

x ¼ A2y ¼ A2

z corresponds to thecubic phase, the state with A2

x ¼ A2y < A2

z etc. to thetetragonal one with c=a < 1 ("3 < 0, "2 ¼ 0, etc.), that withA2x ¼ A2

y > A2z etc. to the tetragonal one with c=a > 1

("3 > 0, "2 ¼ 0, etc.) and that with A2x < A2

y < A2z etc. to

the orthorhombic one ("3 6¼ 0, "2 6¼ 0 etc.). We note that theword ‘‘Triple-Q state’’ is hereinafter used in dual sense. AtT ¼ 0 K, it means the MSDW state with cos � ¼ 1=

ffiffiffi3p

and’ ¼ �=4 in Fig. 2 for example. At finite temperatures belowTN, it means the state where there is no polarization amongQi’s due to the thermal fluctuation. To differentiateexperimentally the non-collinear Triple-Q state from theone where the collinear Single-Qx, Qy, and Qz domains withequal volume exist has been the controversial subject.17–19)

In this respect, the anisotropy measurement of �-ray

emission from spin-polarized nuclei at mK gives theevidence for the existence of the non-collinear Triple-Qstate in an fcc MnNi alloy.20)

According to calculations of the tight binding modellevel,21,22) in the absence of the magnetoelastic coupling(g ¼ 0), the coefficient A of the leading term X2

2 þ X23 in

eq. (8) is positive, and the Triple-Q state with A2x ¼ A2

y ¼ A2z

is the stable MSDW state in nearly half-filled d band.Although the first-principles calculations for the fccphase11,12,16,23) have not confirmed the positive A in �-Mnand Mn1�xAx alloys, we assume the positive A. We then takethe coefficient A as þ1, i.e., the unit of energy.

In Figs. 4(a) and 4(b), we show the most stable latticestructure at the absolute zero temperature in the A0–B planefor C ¼ 0:0 and 0.5, respectively. Figure 4 shows thatthe positive (negative) A0 prefers the cubic (tetragonalwith c=a < 1) phase and the positive B can stabilize thetetragonal phase with c=a > 1. The positive C is found tobe needed to stabilize the orthorhombic phase between thec=a < 1 and c=a > 1 phases. The larger (smaller) C givesthe wider (narrower) orthorhombic region.

Our scenario of the concentration dependence of theposition in the A0–B plane is the following [see Fig. 4(b)].We assume B > 0, and note that g2 is quartic of S. For x ¼ 0,due to the large g2=ð2�Þ, A0 is expected to be negative, ifwe note the first-principles electronic structure calcula-tions.4,12,23) With the increase of x, S is decreased by thedilution effect and reaches �0:75S for x � 0:25, wherethe fcc lattice structure is reported at low temperatures inMn1�xAx alloys. Because of the relation g2 / S4, g2=ð2�Þ isreduced to about one-third as x varies from x ¼ 0 tox � 0:25, if we assume that � is insensitive to x. We may

Fig. 4. The phase diagram at the absolute zero temperature in the A0–B

plane for C ¼ 0:0 (a) and C ¼ 0:5 (b) [see eq. (13)]. The lattice structure

in the shaded area in (b) is not determined within the present calculational

accuracy. As for the arrow from P to Q in (b), see text.

Fig. 3. The variables X2 and X3 take the value inside or on the regular

triangle.

J. Phys. Soc. Jpn., Vol. 78, No. 12 Y. MATSUURA and T. JO

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Page 4: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

therefore expect the positive A0 at x � 0:25, the change fromP (x ¼ 0) to Q (x � 0:25) in Fig. 4(b) and the successivetransition from the tetragonal phase with c=a < 1 to the fccone passing through the orthorhombic and the tetragonalwith c=a > 1 ones with the increase of x, regardless of thedetails of the x-dependence of A.

4. Finite Temperatures and Phase Diagram

We restrict our discussion to the phase transition withinthe antiferromagnetic region (see Fig. 1). As for the para-magnetism-to-antiferromagnetism transition, we refer to aphenomenological discussion.13) We adopt an ordinarycanonical ensemble method for the Hamiltonian given byeq. (7) described in terms of the continuous variables X2 andX3 or Ai (i ¼ x; y; z) with the condition A2

x þ A2y þ A2

z ¼ 1

and the symmetry strains "2 and "3.We calculate the partition function as a function of "2 and

"3 given by, with the use of the Boltzmann constant kB,

zð"2; "3Þ ¼ZA2xþA2

yþA2z¼1

dAx dAy dAz e�H=ðkBTÞ ð14Þ

and the free energy

Fð"2; "3Þ ¼ �kBT ln z; ð15Þ

and determine the stable structure in theffiffiffigp

–kBT plane. Theaverage of a physical quantity X is calculated as

hXi ¼

ZdAx dAy dAz Xe�H=ðkBTÞ

zð"2; "3Þ: ð16Þ

In Fig. 5 we show the obtained phase diagram. With thedecrease of

ffiffiffigp

, i.e., the increase of x, the phase diagramof Mn1�xAx (A ¼ Ni, Ga, Rh, and Au) is found to bereproduced at least qualitatively. Although the orthorhombicphase is extremely narrow at kBT � 0:08, there exists theorthorhombic phase between the two tetragonal phases overthe all range of the temperatures. The magnitude of X2 andX3 is �1, while that of "2 and "3 is �0:01. We assume theg and � values so that Helectron, Hcoupling, and Helastic arecomparable to one another.

In Fig. 6 we show the jump of the strain " (¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi"22 þ "23

p) at

the transition temperature from the cubic phase to the otherones as a function of

ffiffiffigp

. The closer the value offfiffiffigp

approaches to the value at the boundary between the two

tetragonal phases, the smaller the jump of the strain "becomes, and it becomes zero at the boundary point, that is,the cubic to the orthorhombic transition point, where thec=a < 1 and c=a > 1 phases meet, is second order.

The temperature dependence of the symmetry strains"2 and "3 at

ffiffiffigp ¼ 11:2 is shown in Fig. 7. The transition

from the cubic to the tetragonal phase is first order, i.e.,the jump of "3. There exists the jump of the symmetrystrain "2 at the transition temperature from the tetragonal(c=a < 1) phase to the orthorhombic one. We also foundthe jump of "2 at the tetragonal-to-orthorhombic transitiontemperature for

ffiffiffigp ¼ 11:4 and 11.6, but there is no jump of

"2 with varyingffiffiffigp

at T ¼ 0 K. Then the phase transitionfrom the tetragonal (c=a < 1) phase to the orthorhombic oneis first order except for the absolute zero temperature. On theother hand, there is no jump of the symmetry strains at theboundary between the tetragonal (c=a > 1) phase and theorthorhombic one for kBT ¼ 0:0, 0.02, and 0.04. Then thephase transition from the tetragonal (c=a > 1) phase to theorthorhombic one is second order including finite temper-atures.

The temperature dependence of hA2xi, hA2

yi, and hA2z i atffiffiffi

gp ¼ 11:2 is shown in Fig. 8. We note that the Qz

component of the magnetic moment in Mn alloys takeslarger values than those of the Qx and Qy components whenthe lattice contracts along the z axis. All Q components havedifferent values from each other in the orthorhombic phase.Each Q component varies discontinuously at the transitiontemperature from the cubic phase to the tetragonal one with

Fig. 5. The phase diagram in theffiffiffigp

–kBT plane with use of the free

energy [eq. (15)]. Other parameters are taken as B ¼ 0:5, C ¼ 0:5, and

� ¼ 5000:0.

Fig. 6. The jump of the strain at each transition temperature from the

cubic phase to the other ones.

Fig. 7. The temperature dependence of the symmetry strains "2 and "3 atffiffiffigp ¼ 11:2.

J. Phys. Soc. Jpn., Vol. 78, No. 12 Y. MATSUURA and T. JO

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Page 5: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

c=a < 1, and it also varies discontinuously at the transitiontemperature from the tetragonal phase with c=a < 1 to theorthorhombic one. For

ffiffiffigp ¼ 12:8 (9.7), where the ortho-

rhombic phase is not realized at T ¼ 0 K (see Fig. 5), thestable structure is the Single-Q (Double-Q) state only at theabsolute zero temperature.

In the phase diagram (see Fig. 5), the two boundariesbetween the c=a < 1 and c=a > 1 phases and the ortho-rhombic one are remarkably curved, which is found to be acharacteristic property in a wide range of parameter values.This is not reported experimentally, although the experi-mental data points near the second order cubic-to-ortho-rhombic transition in the T–x plane are far from enough.We note the curved boundaries are also reported in thecalculated phase diagram in the JT system.5) In Fig. 5 thereexists the reentrance from the tetragonal phase with c=a > 1

to the cubic one atffiffiffigp � 9:45, but this is not reported

experimentally either.We try taking the values of parameters as B ¼ 2:0, C ¼

5:0, and � ¼ 5000:0. In this case the higher order terms ofHelectron have much larger contribution than the leading termðX2

2 þ X23Þ. The phase diagram is shown in Fig. 9. Compared

to Fig. 5, the orthorhombic phase is wider, and there is noreentrance from the tetragonal phase with c=a > 1 to thecubic one. The tetragonal (c=a > 1) to the orthorhombictransition remains second order, but in contrast to Fig. 5, thetetragonal (c=a < 1) to the orthorhombic transition is secondorder even at finite temperatures. Then the order of the phasetransition is dependent on the values of the parameters.

5. Higher Order Coupling and Elastic Energy

In the previous sections, we took into account only thelowest order terms inHcoupling andHelastic, and found that theterm CðX3

3 � 3X3X22Þ

2 with the positive C in Helectron [seeeq. (8)] is necessary for the orthorhombic phase to appear.This term, which is the 12-th order with respect to Ax, Ay,and Az, corresponds to the 12-spin interaction. In thissection, we examine another origin of the term. If we takeinto account the higher order terms of the coupling be-tween MSDW states and lattice distortion g0½ð"23 � "22ÞX3 �2"2"3X2� and of the elastic energy ð�0=2Þð"22 þ "23Þ

2 instead ofthe term CðX3

3 � 3X3X22Þ

2, the Hamiltonian in this case iswritten as follows

H ¼ Helectron þHcoupling þHelastic ð17Þ

with

Helectron ¼ ðX22 þ X2

3Þ þ BðX33 � 3X3X

22Þ; ð18Þ

Hcoupling ¼ gð"2X2 þ "3X3Þ

þ g0½ð"23 � "22ÞX3 � 2"2"3X2�; ð19Þ

Helastic ¼�

2ð"22 þ "

23Þ þ

�0

2ð"22 þ "

23Þ

2: ð20Þ

By using the conditions @H=@"2 ¼ 0 and @H=@"3 ¼ 0, whichminimize the Hamiltonian H, we obtain the followingequations

gX2 � 2g0ð"2X3 þ "3X2Þ þ �"2 þ 2�0"2ð"22 þ "23Þ ¼ 0; ð21Þ

gX3 � 2g0ð"2X2 � "3X3Þ þ �"3 þ 2�0"3ð"22 þ "23Þ ¼ 0: ð22Þ

With eqs. (21) and (22), we remove "2 and "3 from ourHamiltonian, and obtain the term ðX3

3 � 3X3X22Þ

2 with apositive coefficient. The Hamiltonian (17) with eqs. (18),(19), and (20) is found to give a phase diagram similar toFig. 5. The origin of the term ðX3

3 � 3X3X22Þ

2 is more likelyfrom the higher order coupling and elastic energy than the12-spin interaction.

6. Conclusions and Discussions

In this study, we showed that the phase diagrambelow the Neel temperatures of the first-kind antiferro-magnets Mn1�xAx (A ¼ Ni, Ga, Rh, and Au) is reproducedby using the Hamiltonian which takes into account thecoupling between MSDW states and lattice distortion,and elucidated the relation between MSDWs and latticedistortion.

In our Mn1�xAx alloys, the coefficient of the leadingterm X2

2 þ X23 in eq. (8) is assumed to be positive, and the

Single-Q state described by Qz with the tetragonal (c=a < 1)distortion is realized for small x due to the strongmagnetoelastic coupling term �g2=ð2�Þ. As x (Au etc.)increases, the sublattice magnetization decreases. If weassume the simple dilution effect and the elastic constant � isinsensitive to x, g2=ð2�Þ is reduced to �1=3 from x ¼ 0 tox � 0:25. This stabilizes the Triple-Q state even in T ¼ 0 K.

The transition from the cubic phase to a distorted one isfirst order except at the boundary of the two tetragonalphases (c=a < 1 and c=a > 1). The order of the transitionfrom the tetragonal (c=a < 1 and c=a > 1) phases to theorthorhombic one changes with the values of parameters. Inthe cubic phase, the square of the Q component (A2

x , A2y , and

Fig. 8. The temperature dependence of hA2xi, hA2

yi, and hA2z i at

ffiffiffigp ¼ 11:2.

Fig. 9. The phase diagram in theffiffiffigp

–kBT plane with use of the free

energy [eq. (15)]. Other parameters are taken as B ¼ 2:0, C ¼ 5:0, and

� ¼ 5000:0.

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Page 6: Theory of Multiple Spin Density Wave and Lattice Distortion in fcc Antiferromagnets

A2z ) take the same values, but in the tetragonal phase with

c=a < 1 (c=a > 1), A2z takes larger (smaller) values than

those of A2x and A2

y . In the orthorhombic phase they takedifferent values from each other.

The calculated curved phase boundaries from the twotetragonal phases to the orthorhombic one starting from thecubic-to-orthorhombic second order transition are not yetconfirmed experimentally (see Fig. 5). The detailed meas-urement of concentration dependence may be needed toobtain more detailed phase diagram. To discuss finitetemperature properties in our system, in which electronshave itinerant nature, spin fluctuation theories24) peculiar toitinerant systems might be needed. We expect our work canbe a starting point to such theories.

Finally, we compare our theory with the Kataoka–Kanamori (KK) theory of JT distortion5) in the mixedcrystal of CuCr2O4 and NiCr2O4, which exhibits a similarphase diagram.6) First, the variables in JT system arequantum ones of a single ion with discrete eigenvalues,which are treated in molecular field approximation in KKtheory. In our theory, at first sight, Ai’s and its polynomialX2 and X3 look like macroscopic variables. Our startingpoint for the spin system is, we note, the classical spinHamiltonian, which contains higher orders beyond theHeisenberg term. Then if we restrict our discussion to thesymmetry breaking within the antiferromagnetic state (seeFig. 1), our microscopic state is described in terms ofthe continuous variables Ai’s. Second, in KK theory, thecoefficient of ðX2

2 þ X23Þ term A is zero and the renormalized

A� g2=ð2�Þ is always negative. If we look at Fig. 3, this isfound to lead to the non-cubic structure in the ground state.In our theory, the cubic phase exists even in the ground state,i.e., the positive A [see eq. (8)] due to the four-spininteraction or the so-called band effect is the necessaryingredient.

Acknowledgments

The authors express their thanks to T. Oguchi, H.Shimahara, A. Tanaka, and T. Shishidou for valuablediscussions.

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