Theoretical consideration of magnetic phase ...

Theoretical consideration of magnetic phase formation in MnFeAs y P1− y and Mn2− x Fe x As0.5P0.5 systems in the collective electron model V. I. Valkov, A. V. Golovchan, and D. V. Varyukhin Citation: Low Temperature Physics 38, 386 (2012); doi: 10.1063/1.4705372 View online: http://dx.doi.org/10.1063/1.4705372 View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/38/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electronic mechanism of spontaneous magnetostriction in the layered system Mn2−xFexAs0.5P0.5 Low Temp. Phys. 39, 701 (2013); 10.1063/1.4818632 Magnetic order-order phase transitions in itinerant magnets: Fe2− x Mn x As Low Temp. Phys. 37, 309 (2011); 10.1063/1.3592528 Magnetic and magnetocaloric properties of the alloys Mn 2 − x Fe x P 0.5 As 0.5 ( 0 x 0.5 ) Low Temp. Phys. 35, 786 (2009); 10.1063/1.3253401 Peculiarities of the spontaneous and magnetic-field-induced magnetically ordered phases in alloys of the Mn 2 − x Fe x As 0.5 P 0.5 system Low Temp. Phys. 34, 427 (2008); 10.1063/1.2920080 Magnetic properties and magnetic-entropy change of Mn Fe P 0.5 As 0.5 − x Si x ( x = 0 – 0.3 ) compounds J. Appl. Phys. 99, 08Q105 (2006); 10.1063/1.2158969

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LOW TEMPERATURE PHYSICS

VOLUME 38, NUMBER 5

MAY 2012

Theoretical consideration of magnetic phase formation in MnFeAsyP12y and Mn22x FexAs0.5P0.5 systems in the collective electron model V. I. Valkova) A.A. Galkin Donetsk Physico-Technical Institute, National Academy of Sciences of Ukraine, 72 R. Luxembourg Str., Donetsk 83114, Ukraine

A. V. Golovchan A.A. Galkin Donetsk Physico-Technical Institute, National Academy of Sciences of Ukraine, 72 R. Luxembourg Str., Donetsk 83114, Ukraine and Donetsk National University, Universitetskaya Str., 24, Donetsk 83001, Ukraine

D. V. Varyukhin A.A. Galkin Donetsk Physico-Technical Institute, National Academy of Sciences of Ukraine, 72 R. Luxembourg Str., Donetsk 83114, Ukraine

(Submitted August 11, 2011; revised November 22, 2011) Fiz. Nizk. Temp. 38, 496–506 (May 2012) Experimental magnetic field dependences of magnetization in isostructural systems MnFeAsyP1y (0.2 y 0.66) and Mn2xFexAs0.5P0.5 (0.5 x 1.1) are analyzed by using the results of calculations from the first principles and the model approach. It is shown that the basis of the electronic mechanism of changing the type of magnetic phases in the system Mn2xFexAs0.5P0.5 with cationic substitution is the change in the filling of the d-band. In the system MnFeAsyP1y with anionic substitution the destabilization of the ferromagnetic phase and the occurrence of an antiferromagnetic one with decreasing the arsenic concentration can be caused by a change of the width of density of electronic states, owing to a considerable reduction of the unit-cell volume. C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4705372] V Introduction

The idea of creating refrigeration devices based on the use of the giant magnetocaloric effect gave impetus to the study of a number of systems in which order–order and disorder–order phase transitions induced by a magnetic field are accompanied by significant changes in the temperature of the samples. Iron-manganese pnictides Mn2xFexAsyP1y with a hexagonal crystal structure show promise among those systems. This paper is devoted to the analysis of differences in the formation of spontaneous and induced by a magnetic field magnetically ordered states in systems with anionic MnFeAsyP1y and cationic Mn2xFexAs0.5P0.5 substitutions based on calculations based on the first principles of the spin-polarized electronic structure and the modeling approach. 1. Magnetically ordered phases of the MnFeAsyP12y and Mn22xFexAs0.5P0.5 systems and their stability in a magnetic field

A characteristic feature of magnetically ordered phases of systems with anionic MnFeAsyP1y (Ref. 1) and cationic Mn2xFexAs0.5P0.5 (Refs. 2 and 3) substitutions is the presence of spontaneous ferromagnetism for 0.3 y 0.66 (0.7 x 1.1) and antiferromagnetism for y 0.26 (x < 0.6) within the limits of the respective intervals of arsenic (iron) content. Direct measurements of the magnetic structures by neutron diffraction were conducted by M. Bacmann and coauthors only upon anionic substitution in the MnFeAsyP1y system.4 In the ferromagnetic phase the magnetic moments of manganese MMn and iron MFe atoms are directed along the hexagonal axis c and are 2.6–2.9 lB and 1.0–1.5 lB

depending on y. Since both systems have hexagonal crystal structure (symmetry group P62m D33h ) we can assume that the ferromagnetic (FM) state investigated by the authors of Ref. 4 for the MnFeAsyP1y system is structurally similar to the FM state in the Mn2xFexAs0.5P0.5 system. The antiferromagnetic (AFM) phase, investigated only for the MnFeAsyP1y system, although described by a single vector Q ¼ 2p a ½0; qy ; 0, represents two coexisting types of AFM structures—spiral and sinusoidal.4 A helical coil is formed by magnetic moments of manganese atoms in pyramidal positions (3g), which lie in the ac plane perpendicular to the vector of propagation of structure b*. Their magnitude MMn ¼ 2.41 lB is hardly different from that in the ferromagnetic phase. Magnetic moments of iron atoms in tetrahedral positions (3b) are directed along the hexagonal axis c and form a sinusoidal wave of spin density with amplitude MFe ¼ 0.43 lB, which is more than twice less than the magnitude of MFe in the ferromagnetic phase. The structure of the AFM phase in the Mn2xFexAs0.5P0.5 system for x < 0.6 has not been studied directly. However, significant differences can be noted between the reactions of the AFM states of these systems to the effect of a strong magnetic field. However, for a system with anionic substitution in antiferromagnetic samples of MnFeAs0.25P0.75 and MnFeAs0.26P0.74 the ferromagnetic phase may stabilize with increasing magnetic field strength as a result of irreversible and induced by the magnetic field transition of the first kind AFM–FM (Fig. 1(a)). In the H–T phase diagrams shown here such a transition is characterized by a finite value of the critical field of the appearance of the ferromagnetic phase (l0Hk1(T) > 0 in the entire range of temperatures) and by the absence of a field associated with the

C 2012 American Institute of Physics V 1063-777X/2012/38(5)/9/$32.00 386 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.9.8 On: Sun, 21 Dec 2014 19:35:04

Low Temp. Phys. 38 (5), May 2012

FIG. 1. Field dependences of magnetization in Mn2–xFexAsyP1y systems. MnFeAsyP1y (Refs. 4 and 5) (a); Mn2xFexAs0.5P0.5 (Refs. 7 and 8) (b).

disappearance of the FM phase l0Hk2 for T < T12 ¼ 100 K. In this case the saturation magnetization induced by a strong magnetic field of the state (determined by the value Mexp at y ¼ 0.25, Fig. 1(a)) exceeds the saturation magnetization of the spontaneous FM state (y ¼ 0.5, y ¼ 0.3). Antiferromagnetic phase in a magnetic field is stable for these compounds if cooling of the sample takes place in fields below the first critical field value of l0Hk1 4 T. Inversions of induced order–order transitions are observed only when y 0.2. In this case magnetization of the induced state in a constant magnetic field l0H0 ¼ 20 T > l0Hk1 does not reach the saturation magnetization of a spontaneous ferromagnetic state.5 In the Mn2xFexAs0.5P0.5 system with cationic substitution in the AFM compound Mn1.5Fe0.5As0.5P0.5 (Fig. 1(b)) magnetic field induced AFM–FM transitions are reversible (l0Hk1 > 0, l0Hk2 > 0). In this case the saturation magnetization of the induced phase according to Fig. 1(b) is almost twice less than the maximum value of saturation magnetization of the FM state that is spontaneously realized in Mn1.2Fe0.8As0.5P0.5.7,8 2. Results of calculating the electronic structure of the MnFeAsyP12y and Mn22xFexAs0.5P0.5 systems

To analyze the reasons for qualitative differences in the behavior of the basic characteristics of systems with anionic (MnFeAsyP1y) and cationic (Mn2xFexAs0.5P0.5) substitu-

Valkov, Golovchan, and Varyukhin

387

tions we calculated the electronic structure of both systems in the ferromagnetic and nonmagnetic (NM) states.3 In order to study the electronic structure of Mn2xFexAsyP1y alloys we used the fully relativistic Korringa-Kohn-Rostoker method (the SPRKKR package9). In constructing the crystal potential the atomic sphere approximation was used. Exchange-correlation energy was calculated in the local density approximation without gradient corrections.10 Using the data from these calculations we reduce the electronic state density in the ferromagnetic (MFe = 0, MMn = 0, Mtotal = 0) (Fig. 2) and nonmagnetic (MFe : 0, MMn : 0, Mtotal : 0) phases (Fig. 3). Table I shows the calculated values of the local magnetic moments of manganese and iron atoms, MMn and MFe, and the full magnetic moment of an elementary cell MFM for a series of samples of the MnFeAsyP1y system. Since at low temperatures the ferromagnetic state in samples with y ¼ 0.25 and y ¼ 0.26 can be irreversibly induced by an external magnetic field (Fig. 2), i.e., exist after the end of this exposure, it means that the ordered FM and AFM states are in equilibrium and correspond to the local minima of the thermodynamic potential. Therefore the comparison of electronic characteristics of MnFeAs0.25P0.75 with the corresponding characteristics of samples with only one stable ferromagnetic state (y 0.3) allows us to determine the most significant changes in these characteristics occurring with formation of the AFM state. In Fig. 3(a) electronic state densities of samples with y ¼ 0.25 and y ¼ 0.66 were combined for this, which were calculated for real parameters a(y), c(y) of the crystal cell in the NM phase. Analysis of these plots allowed us to find out the essential features that accompany the trend towards the formation of the AFM phase with decreasing arsenic content. As can be seen in the figure, the reduction of arsenic content leads to a restructuring of the local density of states. For a ferromagnetic alloy with y ¼ 0.66 the Fermi level is near the maximum of the peak C. For an AFM alloy with y ¼ 0.25 there is not only a decrease in the height of the main peaks A and C, but also a minimum near the Fermi level between those peaks. There is also a broadening of the d-band, which correlates with a decrease in cell volume due to the substitution of anions having a large ionic radius (As) with anions of a smaller radius (P).3,11 We also note a light increase in the number of d-electrons in the ferromagnetic NdNM and nonmagnetic NdNM > NdNM phases with decreasing y (Table I). Concentration dependences calculated from the first principles of a total magnetic field in the FM state—MFM(x,y), the number of electrons in the d-zone in the FM and the NM phases— NdFM;NM (x,y), and the width of the d-zone—WNM(x,y), measured at one third the height of the central peak, combined with the experimental dependences Mexp(x,y) for a system with anionic (a) and cationic substitution (b), are shown in Fig. 4(a). For a system with cationic substitution the results of similar calculations are illustrated in Figs. 3(b) and 4(b). It is evident that the decrease in iron content does not lead to a narrowing of the d-band, but is accompanied by a change in the height of the peaks in the density of electronic states. According to Fig. 2(b) this is due to the change in the contribution of d-electrons in the Fe atoms. Note also that there is no qualitative reconstruction of the density of states near the Fermi level, which is to the right of the minimum B for the


388



FIG. 2. Full and partial electronic densities of states in MnFeAs0.25P0.75 and MnFeAs0.5P0.5 in the ferromagnetic state. The vertical line indicates the Fermi level. Positive values correspond to the : (upward) direction of the spin, negative ; (downward).

ferromagnetic (x ¼ 1) and to the left for the antiferromagnetic (x ¼ 0.5) alloys (see inset in Fig. 3(b)). Comparison of Figs. 4(a) and 4(b) also shows that in a system with anionic substitution the values of Mexp(y) and MFM(y) (taking into account the accuracy of experimental data) are in satisfactory agreement. In a system with cationic substitution dependences Mexp(y) and MFM(y) are in agreement only in the range 0.8 x 1.1. The qualitative difference of these curves may indicate the emergence of an energetically more favorable canted structure with a spontaneous magnetization for 0.6 x 0.8, which persists in a wide range of magnetic field strengths (including that for the induced order–order transitions for x ¼ 0.5 (Ref. 7)). The

emergence of intermediate magnetic structure between the FM and AFM states in a system with cationic substitution, and its absence in the system with anionic substitution may be due to qualitative differences in the concentration dependence of the d-band population. In Ref. 12, devoted to the analysis of the phase diagram of the Mn2xFexAs0.5P0.5 system we have shown that changing only the electronic filling of the d-zone in the model while keeping the same form of electronic state density allows us to describe a consistent shift of magnetically ordered phases occurring with changing iron concentration. In the range of values 38.52 NdNM 37.91, which correspond to the values of x from 0.8 to 0.6, the canted structure

FIG. 3. Changing the density of states in the samples of MnFeAsyP1y and Mn2xFexAs0.5P0.5 systems in the nonmagnetic phase with changing arsenic (a) and iron (b) concentration. The vertical lines indicate the corresponding Fermi levels. The insets in detail show the position of the Fermi level. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.9.8 On:

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389

TABLE I. Model parameters (n, De00, J), calculated (MiI,II, N), and experimental (Mi ) values of the magnetic moments M(lB) and the number of d-electrons (N) for the respective systems. M*–values, corresponding to the neutron diffraction data (Ref. 9).

y MFe MFe MMn MMn MFM NdNM NdFM n ¼ NdNM =30 De00 J

0.2 1.4574 0.43(1.45 K) 2.779 2.41(1.45 K) 12.387 39.27 38.97 1.309

x MFeI MMnI MMnII MFM NdNM NdFM n ¼ NdNM =30 De00 J

0.5 1.6051 1.8662 2.8244 13.3505 37.587 37.32635 1.253 0 0.449

0.25 1.45

MnFeAsyP1y 0.275 1.2(1.45 K)

2.81 2.6(1.45 K) 12.44 39.247 38.956 1.308 0.146 0.49 0.6 1.5702 1.7841 2.821 13.1041 37.9105 37.64748 1.264 0 0.453

Mn2xFexAs0.5P0.5 0.7 1.5187 1.725 2.8277 12.9002 38.22563 37.95856 1.274 0 0.46

0.3 1.45 1.24(1.45 K) 2.82 2.55(1.45 K) 12.5 39.23 38.94 1.3077

0.5 1.48 1.48(200 K) 2.84 2.02(200 K) 12.68 39.17 38.87 1.305 0 0.476

0.8 1.4887 1.73 2.8621 12.86 38.52 38.24321 1.284 0 0.465

0.9

12.665 38.849 38.557 1.295 0 0.47

0.66 1.49 2.87 12.81 39.12 38.824 1.304 0.094 0.456 1.0 1.4223 2.8621 12.4964 39.16 38.86775 1.305 0 0.476

FIG. 4. Dependences of experimental Mexp and calculated from the first principles WNM, NdNM, NdFM, MFM characteristics of the MnFeAsyP1y and Mn2xFexAs0.5P0.5 systems on arsenic (a) and iron (b) concentration. WNM, NdNM, NdFM, and MFM are the width of the d-band in the nonmagnetic phase (MFM ¼ 0), the number of electrons in the nonmagnetic and ferromagnetic phases, and the calculated magnetic moment per cell; the values of Mexp (~), Mexp (), and Mexp (n) were taken from Refs. 5, 6, and 14, Mexp (*)—from the construction of M(T) in Ref. 5 for H ¼ 20 T; the values of Mexp (~) were taken from Ref. 8 for T ¼ 5 K, the values of Mexp (D) correspond to saturation magnetization extrapolated to zero field for T ¼ 19 K for the measurement of induced order–order transitions in Ref. 7. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.9.8 On:

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390



system can be compared with the formation of a minimum in the density of electronic states DOS(E) near the Fermi level and a relatively strong broadening of the DOS(E) functions in the nonmagnetic phase ½DWNMN =WNM ¼ WNM ½ðy ¼ 0:2Þ WNM ðx ¼ 0:66Þ= WNM ðx ¼ 0:2ÞWNM ¼ 7% (Fig. 4(a)). A similar change in the width of the zone in a system with cationic substitutions Mn2xFexAs0.5P0.5 with decreasing iron content, DWNMN =WNM ¼ ½WNM ðx ¼ 0:4Þ WNM ðx ¼ 1:2Þ=WNM ðx ¼ 0:4Þ; does not exceed 1% (Fig. 4(b)). At the present time a partial confirmation of these assumptions can be illustrated by the single-band model, which describes the magnetization of the spiral structure in the system with itinerant d-electrons.13 3. Induced magnetic phase transitions of the order–order type in the single band model of the spiral structure

In this section the description of a complex magnetic structure of an itinerant magnet will be held in the framework of the single band model, in which delocalized electronic states are characterized by the transfer integrals, and spin polarization is taken into account by introducing the singlecenter exchange interaction. It is assumed that the characteristics of magnetically ordered phases of the Mn2xFexAs1yPy system can be qualitatively taken into account by using the model electron density of states G(E), similar to the shape of the density of electronic d-states DOSd (E), calculated from the first principles for a specific material in the nonmagnetic phase. The magnitude of a single-center exchange integral J, which, like the number of d-electrons in the n state, is in line with the results of ab initio calculations. The model approach allows us to investigate the reaction of the system to a magnetic field, and the change of the function G(E) and parameters J and n. When choosing the direction of the Z axis along the wave vector of the helix, the spiral structure can be described by the Hamiltonian (1), which takes into account the spin polarization of the pie in the basal xy plane in the presence of external homogeneous Hjz ¼ H0 and periodic Hjx ¼ HQ cos(QRj), Hjy ¼ HQ sin(QRj) fields,

FIG. 5. Seed functions g11(e1) and g00(e) and model densities of states in the nonmagnetic (NM) and ferromagnetic (FM) phases. Parameter De00 shows the degree of broadening of the base of the functions g00 (e, y, x) compared with the “standard” g00 (e, y ¼ 0.5, x ¼ 1.0).

emerges. Spontaneous stabilization of this structure in the specified concentration range is, according to Ref. 8, the reason for the fundamental differences between the dependences MFM(x) and Mexp(x) (Fig. 5(b), curves III). In this case, as shown in Refs. 7, 12, and 8, the canted structure retains stability in the experimentally attainable range of magnetic field strengths. Therefore, we can assume that the existence of only the FM and AFM phases in the MnFeAsyP1y system is due to the relatively large permanent value of the electronic filling of the d-band. This leads to a lack of intermediate magnetic structure and to a satisfactory agreement between experimental and calculated first principle values of the FM component of the magnetic moment (Fig. 4(a), curve III). Then the tendency towards the emergence of the AFM state in this

_

_

H ¼ H 0 þ H^ int þ H^ ex

X X r¼þ;

S Q ¼

X k

aþ k# akþQ" ;

S Q ¼

ek a þ kr akr

J

jS2ja j þ glB

X

H0 Sjz þ Hjx Sjx þ Hyx Sjy

j

X

jS2qa j

þ glB ½H0 S0z þ HQ ðSþ Q þ SQ Þ;

(1)

q;a¼x;y;z

k

X

X j;a

k

X X r¼þ;

ek aþ kr akr J

aþ kþQ# ak" ;

Sþ Q ¼

k

The first term ðH^ 0 Þ represents the kinetic energy of the system of non-interacting electrons; the second term ðH^ int Þ describes the exchange energy at the centers, and the third

X k

aþ k" akþQ# ;

S0z ¼

1X þ ða ak" aþ k# ak# Þ: 2 k k"

ðH^ ex Þ describes the energy of interaction with external fields; HQ, ak" ; akþQ# ; are Fourier components of the periodical fields and annihilation operators of electrons in spin-up and




spin-down states, respectively, J is the effective intra-atomic exchange integral; ek are Fourier components of the transfer integrals. Using the method of functional integration, the expres~ g)/N0 of sion for the thermodynamic potential F(n, g) ¼ F(n, a system, described by the Hamiltonian (2), at T ¼ 0 can be reduced to the form,13 Fðn; gÞ ¼ Eðn; gÞ þ Jðn glB H0 =2JÞ2 þ Jðg glB HQ =2JÞ2 ;

(2)

where N0 is the number of magnetically active atoms; E(n, g) is the dependent on n energy of spin-polarized electrons, which is calculated by taking into account the structural characteristics of the model density of electronic states by introducing functions g0(e) and g1(e) Ref. 3, X ð dede1 Em ðe; e1 ; n; gÞ Eðn; gÞ ¼ m¼1;2

H l Em ðe; e1 ; n; gÞ g0 ðeÞg1 ðe1 Þ;

E1;2 ðe; e1 ; n; gÞ ¼ e6

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½e1 Jn2 þ J 2 g2 ;

(3)

2 ð X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1;2 ðk; n; gÞ¼ðe þ ekþQ Þ 26 ½ðek ekþQ Þ=2 Jn2 þ J 2 g2 ;

(5) ð ek þ ekXY þQ ; g00 ðeÞ ¼ dkXY d e XY 2

(6)

ð ek ekZ þQ g11 ðe1 Þ ¼ dkZ d e1 Z : 2

(7)

The relationship between the normalized model electron state density G(E, n, g) and functions g00(e) and g11(e) is given by GðE; n; gÞ ¼

Xð

dede1 d E En ðe; e1 ; n; gÞ g0 ðeÞg1 ðe1 Þ;

n21;2

.ð

go ðeÞ ¼ g00 ðeÞ

deg00 ðeÞ; ð . g1 ðe1 Þ ¼ g11 ðe1 Þ de1 g11 ðe1 Þ:

(8)

(4)

where g0(e) and g1(e) are associated with the main characteristics of the model electronic spectrum Eq. (5) by Eqs. (6) and (7),

H0 =J ¼ 2n þ

391

Solutions of equations of state @F/@n ¼ 0, @F/@g ¼ 0, supplemented by the equation for the chemical potential Eq. (9), are searched for at constant J, n, g00(e) and g11(e1),

. h i g1 ðe1 Þg0 ðeÞ@Em ðe; e1 ; n; gÞ @n H l Em ðe; e1 ; n; gÞ dede1 ;

(9a)

m¼1

0 ¼ 2g þ

2 ð X m¼1

. h i g1 ðe1 Þg0 ðeÞ@Em ðe; e1 ; n; gÞ @g H l Em ðe; e1 ; n; gÞ dede1 ; ð

n ¼ g1 ðe1 Þg0 ðeÞ

2 h i X H l Em ðe; e1 ; n; gÞ dede1 :

(9b) (9c)

m¼1

The relationship between the static components ~ g ¼ gq¼Q of the spin fields with the average n ¼ nq¼0,z, ~ values of spatially homogeneous (q ¼ 0) ferromagnetic ~ 0 Þ and helix amplitude ðglB hSþ moment ðglB hSz i¼ m q¼Q i ~ Q Þ is determined by the ¼ glB hSq¼Q i ¼ glB SQ ¼ m expressions

gl H0 gl HQ ~ Q ¼ glB ~g B ~ 0 ¼ glB ~ ; m : (10) n B m 2J 2J These expressions allow us to compare the solutions to the system of equations (9) n, g ¼ g(n) with, according to Eq. (10), the values m0 ¼ m0(n), mQ ¼ m0(g) with experimentally measured field dependences of magnetization.

3.1. Selection of basic parameters of the model

An important parameter of the model, comparable to the data of ab initio calculations, is the number of electrons in the n(x, y) zone,

n nðx; yÞ ¼ 3NdNM ðMn3f Þð1 xÞ þ 3NdNM ðFe3f Þx þ 3NdNM ðMn3g Þ þ 2ð1 yÞNdNM ðP2c Þ NM þ 2yN NM d ðAs2c Þ þ ð1 yÞNd ðP1b Þ o. 30: þ yN NM d ðAs1b Þ

(11)

In Eq. (11) the first two terms in curly brackets correspond to the number of d-electrons per manganese and iron in tetrahedral (3f) positions of the crystal cell; the third term corresponds to the number of d-electrons in pyramidal (3g) positions of manganese; the fourth and fifth terms—to the number of electrons in the As/P positions with 2c symmetry; the sixth and seventh—to the number of d-electrons that are compatible with the As/P positions with 1b symmetry. All these values were determined directly from the results of calculations based on the first principles (Table I). Thus, Eq. (11) relates the number of electrons per single state in our model to the total number of electrons NdNM (x, Y) found in 30 d-states in


392



the crystal cell of a particular alloy Mn2–xFexAs1yPy in the NM phase (with the same electron filling of the spin up (down) bands). Through selection of functions g00(e) and g11(e) the model density of electronic states GNM(E) ¼ G (E (n ¼ g ¼ 0)) (Fig. 5) can be approximated in form to that obtained from the ab initio calculations of 1/30 of the density of electronic states DOSdNM(E) in the NM phase (Fig. 3). The introduction of these two functions creates a more realistic description of the prerequisites for the model of magnetically ordered states in the Mn2xFexAs1yPy system compared to the model approach in our work,7 where only function g11(e) was considered. In the approach in Ref. 7 only the centrosymmetric electron density of states had to be used. The exchange integral J(x, y) is estimated from expression, l FM DEFM ðx; yÞ ¼ 2JhSz i ¼ J~ d ðx; yÞ=lB ;

(12)

where the exchange splitting DEFM is determined from the displacement of electron state densities DOSUP(E) and DOSDOWN(E) in the FM phase. The assessment of this bias in MnFeAs0.5P0.5 gives values within 0.13(0.20) Ry. The first value is identified with the distance between the centers of the spin subbands at a height of 1/3 of each of the functions DOSUP(E) (DOSDOWN(E)). The second corresponds to the distance between the central peaks DOSUP(E) (DOSDOWN (E)). For MnFeAs0.5P0.5 (considering that the average ferromagnetic moment for a single d-state is equal to ~ FM l d ðx ¼ 1; y ¼ 0:5Þ ¼ MFM ðx ¼ 1; y ¼ 0:5Þ=30 ¼ 12:91lB =30 ¼ 0:43lB Þ we get J ¼ 0.285(0.476) Ry. In the model calculations we used the value J ¼ 0.476 Ry, which leads to rigid displace-

ment of model densities of electronic states GUP(E) and GDOWN(E) in the FM phase by 0.15 Ry (Fig. 5).

3.2. Analysis of the solutions of equations of state

Equations of state (9) were solved for the set of seed functions g00(e) and g11(e) that determine the normalized model density of electronic states (Fig. 5) and for the specified number of d-electrons n that agrees with the chemical composition. In the investigated range of concentrations of cations 0.5 x 1.0 for y ¼ 0.5 and anions (0.25 y 0.66) for x ¼ 1.0 the change in the concentration of electrons n, according to Eq. (11), is uniquely determined by the interval 1.253 n 1.308. For each value of J, n, and seed functions g00(e) and g11(e) there are two types of solutions that determine in Figs. 6 and 7 the corresponding energy minima when H ¼ 0. Values marked mFM are the solution of equations when g : 0. These solutions exist for any values of n(x, y) and correspond to the ferromagnetic state that is calculated from the first principles for the investigated range of concentrations and forms the dependence MFM(x, y) (Fig. 4). The second solution, marked in Figs. 6 and 7 as m0, is related to removing the limitation g : 0. This solution can describe either the ferromagnetic state for m0 ¼ mFM, mQ ¼ 0, or the canted state for 0 < m0 < mFM, jmQj > 0. As can be seen in Fig. 6, the simulation of a system with anionic substitutions, in which the reduction of arsenic content leads to a significant broadening of the d-band WNM, occurs by increasing the parameter De00(y). In this case, in lieu of the almost constant value of the parameter n 1.307 simultaneous coexistence of values m0 and mQ(m0) for H ¼ 0 is not observed. This

FIG. 7. Model change in the energy stability of magnetically ordered states FIG. 6. Model change in the energy stability of magnetically ordered states with changing the electronic filling of zone n and De00 ¼ 0. x ¼ 0.5 with changing the parameter De00. y ¼ 0.25 (n ¼ 1.308) (a); y ¼ 0.5 (n ¼ 1.253) (a); x ¼ 0.6 (n ¼ 1.264) (b); x ¼ 1.0 (n ¼ 1.305) (c). (n ¼ 1.305) (b); y ¼ 0.66 (n ¼ 1.304) (c). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.248.9.8 On:

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FIG. 8. Model description of the magnetic field induced order–order transitions with respect to alloys MnFeAs0.25P0.75 (a) and Mn1.5Fe0.5As0.5P0.5 (b). The inset in (a) shows the initial densities of electronic states used in the calculation of magnetization curves.

means that the antiferromagnetic contribution mQ to the total magnetic moment upon magnetization equals zero. A qualitatively different situation is observed for significant changes of the number of electrons n at a constant width of electron state density (De00(x) ¼ 0) (Fig. 7). This situation is typical for a system with cationic substitutions (Fig. 4(b)). As seen in Fig. 7, here, along with the ferromagnetic state (m0 ¼ mFM, jmQj > 0) for n 1.305, with a decreasing n (iron content) the canted state with nonzero values of ferromagnetic m0 < mFM and antiferromagnetic jmQj > 0 components of the magnetic moment can also become stabilized. For example, for n ¼ 1.264 the deepest energy minimum corresponds to the canted state with coexisting ferromagnetic (m0) and antiferromagnetic (mQ) contributions. Such a case seems to be realized in the Mn2xFexAs0.5P0.5 system when 0.5 < x < 0.8, in particular for x ¼ 0.6 (n ¼ 1.264). Antiferromagnetic state (m0 ¼ 0, jmQj > 0) is realized when n ¼ 1.253 (De00 ¼ 0) for a system with cationic substitutions (x ¼ 0.5) (Fig. 7(a)), and when n ¼ 1.308 (De00 ¼ 0.146) for a system with anionic substitutions (y ¼ 0.25) (Fig. 6(a)). As shown by model calculations, in the first case the minimum energy at m0 ¼ 0, jmQj > 0 is due to lower electronic band filling in n. In the latter case for large values of n the AFM state is due to the large value of the parameter De00, which in the model is compared with the broadening of the d-band WNM. At the same time, the minima for the AFM and FM states are separated by a potential barrier. This is a special case in which each of these states may persist in the absence of the field. In Fig. 8 both of these cases are presented in the form of field dependences of magnetization, which simulate irreversible (Fig. 8(a)) and reversible (Fig. 8(b)) order–order transitions observed in both systems experimentally (Fig. 1).

As seen in Fig. 8, with increasing magnetic field the dependences m0(H) (curve 2) and mQ(H) (curve 1) describe two qualitatively different order–order transitions of the first kind. In Fig. 8, while increasing the magnetic field strength, a gradual increase of the m0 component, accompanied by a decrease in the mQ component, is disrupted when the value of the first critical field H1 is achieved. In this field an irreversible AFM–FM transition of the first kind is realized, which results in an abrupt increase of the m0 component up to the value of mFM and complete disappearance of the mQ component. Reverse transition does not occur with decreasing magnetic field strength. Magnetization curves in Fig. 8(a) model the experimentally observed irreversible transition of the first kind that is realized in MnFeAs0.25P0.75 (Fig. 1(a)). With increasing magnetic field strength similar dependences in Fig. 8(b) demonstrate the transitions of the first kind from antiferromagnetic to canted state (AFM–CS), in which the parameter m0 although increases sharply does not reach the value of mFM, and the change in the component mQ does not lead to its complete disappearance. This transition is reversible, since it is characterized by the presence of two critical fields H1 and H2, and at lower magnetic field strength the AFM state is restored. These curves simulate a reversible transition of the first kind, observed in the system with cationic substitutions in the Mn1.5Fe0.5As0.5P0.5 sample (Fig. 1(b)). Note also that for n ¼ 1.309 and an even larger increase in the parameter De00 magnetization of the spiral structure is realized as a reversible AFM–CS transition. In this case there is a significant decrease in the saturation magnetization of the induced phase. This model result does not go against experimental data, characteristic for the samples of the MnFeAsyP1y system with arsenic content below 0.25.6


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Conclusion

The change in the type of magnetically ordered state in a system with cationic substitutions Mn2xFexAs0.5P0.5 with decreasing iron content is due to a significant decrease in the electron filling of the d-band. Between the ferromagnetic (0.8 x 1.1) and antiferromagnetic (x < 0.6) states in a system with cationic substitution Mn2xFexAs0.5P0.5 an intermediate phase emerges (0.6 x 0.8), which exhibits spontaneous magnetization and maintains stability in a magnetic field. The appearance of this phase is associated with an intermediate electronic population of the d-band manifested with decreasing iron concentration. In a system with anionic substitutions MnFeAsyP1y when 0.25 y 0.66 the ferromagnetic and antiferromagnetic states are the closest in energy competing phases. Stabilization of the AFM phase with decreasing arsenic concentration while retaining the population of the d-band may be due to the broadening density of states, which is due to a decrease in the unit cell volume. The work was performed as part of a competitive project DFFDU-BRFFD No. 41.1/038. a)

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1

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Translated by D. K. Maraoulaite
