Axial quadrupole phase of a uniaxial spin-1 magnet

Author’s Accepted Manuscript Axial quadrupole phase of a uniaxial spin-1 magnet P.A. Sayko, I.P. Shapovalov

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S0304-8853(15)30112-8 http://dx.doi.org/10.1016/j.jmmm.2015.04.117 MAGMA60206

To appear in: Journal of Magnetism and Magnetic Materials Received date: 3 February 2015 Revised date: 20 April 2015 Accepted date: 26 April 2015 Cite this article as: P.A. Sayko and I.P. Shapovalov, Axial quadrupole phase of a uniaxial spin-1 magnet, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2015.04.117 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Axial quadrupole phase of a uniaxial spin-1 magnet P.A. Sayko1, I.P. Shapovalov2 I.I.Mechnikov Odessa National University, 2 Dvoryanskaya str, Odessa 65082, Ukraine 1

Corresponding author, e-mail: [email protected]

2

E-mail: [email protected]

ABSTRACT The axial quadrupole phase of uniaxial spin-1 magnet in an external magnetic field has been investigated. The case of magnetic system with the most general form of single-ion anisotropy and anisotropic biquadratic exchange interaction is considered. It is shown that the relative magnetization in the molecular field approximation does not depend on temperature and linearly increases with external magnetic field. Two branches of the spin excitation spectrum are determined. The boundary between the axial quadrupole and angular phases is defined by the condition for softening of the spectrum. The critical temperature of the corresponding phase transition considerably depends on the anisotropy constants of the biquadratic exchange interaction. Keywords: Uniaxial magnets; Single-ion anisotropy; Biquadratic exchange interaction; Quadrupole phase. 1. Introduction Investigation of strong magnets with spin ½ is traditionally based on Heisenberg model. The Hamiltonian of this model includes exchange interaction (EI) energy and Zeeman energy. In the magnetic systems with unity spin the tensor interactions such as the single-ion anisotropy (SIA) and the biquadratic exchange interaction (BEI) can also be present along with the two interactions considered above. There are many magnetic compounds for which the tensor interactions constants comparable with the EI constant [1]. To describe the properties of such substances adequately, Heisenberg model should be expanded by including SIA and BEI. The presence of tensor interactions may cause quadrupole ordering of magnetic moments in the system. A characteristic feature of the quadrupole ordering is in the fact that at zero magnetic field value the magnetization of the system equals zero, but the quadrupole magnetic moment has a non-zero average value [1]. There are two different types of quadrupole phases in the systems with unity spin. The first type corresponds to the condition which in absence of the external magnetic field at zero temperature has the form:  S    0 , (S  )2   1 , where S  is the

projection of a magnetic moment of ion onto the ordering axis α [1]. Phases of such type will be denoted as QP1α. In these phases S  is equal 1 or -1 with the same probability, i.e. phases of QP1α type are the axial quadrupole phases. The second type of quadrupole ordering corresponds to the condition   S   0 , (S  )2   0 [1]. Phases of such ordered type will be denoted as QP2α. The magnetic moments of ions in these phases are freely rotating in the plane normal to the axis α. In other words, the phases of QP2α type are the planar quadrupole phases. There are many scientific publications dedicated to studying of the properties of the magnetic systems with SIA and isotropic BEI [2-11]. But magnets with anisotropic BEI are studied insufficiently. It should be noted, that the few works which take BEI anisotropy into account show that the change in the anisotropy constants can lead to a significant change in the magnetic properties of the system [12-15]. Main purpose of this paper is the study of QP1Z -phase (where Z – symmetry axis) for spin-1 magnet with the most general form of anisotropic tensor interactions in presence of the longitudinal external magnetic field. 2. Temperature and field dependence of the order parameter The most general form of the Hamiltonian for the uniaxial spin-1 magnet with SIA and BEI in a longitudinal magnetic field is given by [12]: H  h

S i

Z i







1   J ij S iZ S Zj   S iX S jX  S iY S Yj  D  Qi0  3 i, j i



 



1    K ij  Qi0 Q 0j   Qi1Q1j  Qi1Q j 1   Qi2 Q 2j  Qi 2 Q j 2  , 3  i, j

(1)

where h is the strength of the external magnetic field directed along the OZ-axis; J ij are the EI constants;  is EI anisotropy constant; D is SIA constant ; K ij are BEI constants;  ,  are BEI anisotropy constant; S l ( l  x, y, z ) are spin operators; Q p ( p  0,1,2 ) are quadrupole operators; and i , j enumerates the indices of the crystal lattice sites. Relation between quadrupole ( Q p ) and the spin operators are determined by the formulas: 2 Q 0  3S z   2I , Q1  S z S y  S y S z , Q 1  S z S x  S x S z ,

   S  ,

Q2  S x

2

y 2





Q 2   S x S y  S y S x ,

where I is the unity matrix. The matrix form of the operators S l and Q p is given in the Appendix. We consider the case of the system with only one-sublattice phases. It is possible under the conditions J ij  0 and K ij  0 . In this case the average values of

 S l  and Q p  fully determine the magnetic structure of the system and can be used

as the order parameter (OP) components. There are only two non-zero components of the OP: Q 0  and Q 2  in QP1Z –phase within the absence of an external magnetic field [14]. Presence of the external magnetic field leads to the appearance of the non-zero longitudinal component of magnetization:  S z   0 . The corresponding Hamiltonian of the system in the mean-field approximation takes the form of





H 0   h  2 J 0  S Z   S iZ  i





1 D  2 K 0 Q 0   Qi0  2K 0 Q 2   Qi2 , 3 i i

(2)

where J 0 and K 0 are defined by the equations J 0   J ij , K 0   K ij . i

i

Diagonalization of an arbitrary single-particle Hamiltonian of the spin-1 magnet may be performed by a unitary transformation of operators S l and Q p : ~ ~ S l  VS lV 1 ; Q P  VQ PV 1 , (3) ~l ~p where S and Q are spin and quadrupole operators in the new basis, and V is a suitable unitary transformation. General form of the transformation was obtained in [12]: V  exp  i Q 2  exp i Q1  exp i S y  , (4) where  ,  , are the transformation parameters. For Hamiltonian (2) the transformation parameters may be defined as follows:     2 ,   4 , sin 2   ~

h , K 0  J 0    

(5)

~

where    S Z  ,   Q 0  . Values of  and  depend on the Hamiltonian parameters and the temperature. By averaging the relation (3), and taking into account the explicit form of the transformation V , we can find the non-zero OP components as a function of the Hamiltonian parameters and values of  and  :  S  Z

h

2K 0  J 0 

, Q   3    2 , Q    0

2

K 0   J 0 2    2  h 2 . (6) 2K 0   J 0 

From the first equation (6) we have the following interesting result: in QP1Z phase the relative magnetization within the mean-field approximation does not depend on temperature and linearly increases as the external magnetic field. In the new basis Hamiltonian H 0 is diagonal: ~ ~ H 0  ~1  S iZ  ~2  Qi0 (7) i

i

where values of ~1 and ~2 are defined by the following expressions:

~1  D  2K 0  K 0   1    , 1 2

1 2

~2   D  K 0  K 0   1    . 1 6

2 3

1 2

(8)

The energy of a single ion with different values of S~ z ( S~ z  1, 0,1 ) is determined by the formulas: (9) E1  ~1  ~2 , E0  2~2 , E1  ~1  ~2 . The values of  and  may be determined by Gibbs measure:

 S exp  E kT  ,   exp  E kT  ~Z n

n

n

n

n

 Q exp  E kT  ,   exp  E kT  ~Z n

n

n

(10)

n

n

where k is the Boltzmann constant, and T is the absolute temperature. We consider the basis defined by (5) as a new basis of the operators. Then, the solution of the system (10), which corresponds to QP1Z –phase, satisfies the following condition: if T→ 0 then   0,   2 . Thus, the lowest energy level of ~ ion is the level with S Z  0 . It follows from formulas (8-10) that the values of ~1 and ~2 , the energy levels of a single ion E n , and the values of  and  do not depend on the external magnetic field. 〈Q0〉

1.0

0.5

0.0

1.0

2.0

3.0

𝜃С 𝜃

Fig.1. Temperature dependence of the quadrupole magnetization Q 0  at J0=1.0; ξ=1.0; D=1.2; K0=1.25; η=2.0; ζ =3.0.

The field independence of  and  leads to the fact that in the QP1Z -phase OP quadrupole component Q 0  which is defined by the second formula (6) does not depend on the value of h. The dependence of the Q 0  from dimensionless ~ ~ temperature     J 0  is shown in Fig.1. It should be noted, that with increasing field the phase transition (PT) of the second kind of phase QP1Z in the ~ ferromagnetic phase (FMP) takes place [13,16]. The critical temperature  С of this

phase transition, i.e. point where line Q 0   f   ends, depends on the h value. In ~ Fig.1 the value  С corresponds to the value of the dimensionless magnetic field ~ ~ h  4,5 h  h J 0 . ~

〈Q2〉

𝜃С3 𝜃С2 𝜃С1 0.0

2.0

1.0

𝜃

3.0

3 -0.2

2 -0.4

1 -0.6

Fig.2. Temperature dependence of the quadrupole magnetization Q 2  at different values ~ ~ of dimensionless field: curve 1 refers to h  4.7 , curve 2 refers to h  5.0 , curve 3 refers to ~ h  5.3 and J0=1.0; ξ=1.0; D=1.2; K0=1.25; η=2.0; ζ =3.0.

The dependence of Q 2  from the dimensionless temperature at different values of the external magnetic field is presented in Fig.2. Since the FMP satisfies ~ the condition Q 2   0 , the points  Cm ( m  1, 2, 3 ) for each curve in Fig.2 are the points of the phase transition from QP1Z-phase into FMP. Thus, the family of curves in Fig.2 allow us to define a set of points of the phase boundary between ~ ~ QP1Z and FMP in coordinates h   . 𝜃 4.0

3.0

2.0

QP1Z

FМP

1.0

0.0

1.0

2. 0

3.0

4.0

5.0

6.0

7.0

ℎ

Fig.3. The boundary between QP1Z-phase and FMP at J0=1.0; ξ=1.0; D=1.2; K0=1.25; η=2.0; ζ=3.0.

This boundary is shown in Fig.3. It fully coincides with the line where the spin excitations spectrum in FMP, which are built by the algorithm of work [16], loses it stability. 3. Spin excitation spectrum ~

Since in the new basis the lowest energy level of a single ion has S Z  0 , the ~ ~ states with S Z  1 and S Z  1 are excited. Two branches of the spin excitation spectrum under the condition of absence of the external field - h  0 were obtained in [14]: 1 k   2    K k  K 0      K 0   1  D

(11)

 2    J k  K 0      K 0   1  D , 12

2 k   2     K 0  K k  K 0  J k  .

(12) In the long-wave limit, i.e. at condition k  0 , the 1 (k ) branch corresponds to quadratic dispersion law, and the  2 (k ) branch corresponds to linear dispersion law. Generalization of the calculations for the case when an external magnetic field has a non-zero value gives us the following expression for the first branch of the spectrum: 12 , (13) 1 (k )  Lk   L2 (k )  4M (k ) where L(k )  Ak2  Bk2  Gk2  H k2  2Ck Ek  2Dk Fk ,





Ak M (k )  det

Bk

 Bk  Ak Ek  Fk

Fk

Ck

Dk

 Dk  Ck Gk

Hk

,

 E k  H k  Gk





Ak  h sin 2   D  3    K 0      J 0 sin 2 2    K 0 cos 2 2  

     J k   K k ,

Bk      cos 2  J k   K k  ,

Ck  h cos 2     sin 2 cos 2 J 0   K 0  , Dk     sin 2  K k   J k  , Ek  Ck , Fk  2 sin 2  K k   J k  ,





Gk  h sin 2   D  3    K 0      J 0 sin 2 2    K 0 cos 2 2    2   J k   K k ,

H k  2 cos 2  K k   J k  .

In the long-wave limit the dispersion law (13), as well as (11) is the quadratic one. The condition of the spectrum stability for the 1 k  branch has the form 1 (0)  0 , and stability boundary is: (14) 1 (0)  0 . The equation (14) is equivalent to the following system: (15) L(0)  0 , (16) M (0)  0 . The second branch of the spectrum has the form  2 (k )  2{h sin 2   (   )(K k  J 0 ) sin 2 2   (   )(K 0  J k ) cos 2 2  ] 



(17)



 h sin 2   (   )(K k  J 0 ) sin 2 2    (   )( K 0  K k ) cos 2 2  }1 2 .

This branch of the spectrum at condition k  0 has the quadratic dispersion law. Thus, if we switch on the magnetic field, the dispersion law for  2 (k ) branch changes from the linear form to the quadratic one. At zero temperature, the expressions (13) and (17) are fully consistent with the results of work [12], and at finite temperatures and zero field value they are fully consistent with the results of work [14]. It should be noted, that at condition h  0 the mode of the  2 k  branch of the spectrum with zero wave vector is the Goldstone mode. For finite values of the field this property disappears. 𝜃∗

𝜃∗

Место для формулы.

4.0

1

3.0

1

2

2 3.0

2.0

3

3

1.0

2.0

2.5

3.0

ζ

3.5

2.0 1.0

2.0

3.0

4.0

η

5.0

~ Fig.4. Dependence of the dimensionless critical temperature  * on ζ at constant value of ~ η=2.0 and  * on η at constant value of ζ=3.0 for different values of dimensionless field: curve 1

~ ~ ~ refers to h  2.5 , curve 2 refers to h  3.2 , curve 3 refers to h  3.8 and at J0=1.0; ξ=1.0; D=1.2; K0=1.25.

The condition (14) determines the line of second-order phase transitions between QP1Z and angular phases. In angular phase the magnetization makes certain angle with the symmetry axis OZ [12, 13]. The critical temperature ~ ~  * (  *   * J 0 ) of this phase transitions is an implicit function of the BEI anisotropy constants  and  . The dependence of the dimensionless critical ~ temperature  * on the values  and  at different values of the dimensionless field ~ h is shown in Fig.4. 4. Discussion of results It is shown above, that the ground state energy in QP1Z –phase within meanfield approximation does not depend on value of external magnetic field. The same property is valid for QP2Z –phase [12]. So, the phase transitions (PT) between axial and planar quadrupole phases at zero temperature are impossible. Thus, if there is a QP1Z-phase at zero temperature, and there is no angular phase realized in the system, then QP1Z –phase will be conserved with decreasing of field value until it reaches h  0 . If the angular phase can be realized in the system, then PT temperature from QP1Z-phase into angular one strongly depends on BEI anisotropy constants  and  . This fact speaks in favor of such models of the uniaxial spin-1 magnet which take into account BEI in the anisotropic form. Besides, such effect as quadrupole hysteresis can be studied only with taking into account BEI anisotropy [15]. With increasing of magnetic field h the second order phase transition from QP1Z –phase to FMP inevitably occurs. The boundary between these phases in the field - temperature coordinates is the stability boundary of the spin excitations spectrum in the FMP [16]. Thus, the definition of the magnon spectrum in the FMP is the standard way to find the line of the second order PT between phases QP1Z  FMF. In Section 2 we propose the alternative method for determining of the phase boundary. This method uses the fact that the order parameter component Q 2  in the FMP equals zero, and in QP1Z -phase it is non-zero. We suppose, that for experimental check of this work results it is advisable to use the magnetic compound, for which BEI plays determinative role in the magnetic ordering. Suitable magnet can be found in the class of RT2X2 , where R is a rare-earth element, T is transitional metal, and X is Ge or Si. Such magnets were studied in works [17,18].

Appendix The matrix form of S l and Q p operators: 1 S  0 0 Z

0 0 0 0 , 0 1

1 0 Q  0 2 0 0

0 0 , 1

0 Q2  0 1

1 0 , 0

0

0 0 0

S

X

0 1  1 2 0

0 Q  1 2 0 1

Q

2

i

0 i 0 1

1 0 1

0 1, 0

1 0 0 1 , 1 0

0 0 0

0 S  1 2 0 Y

Q

1

i

1 0 0 1 , 1 0

0 1 0  1 0 1 , 2 0 1 0 1

1 0 . 0

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Highlights ● Quadrupole phase of uniaxial spin-1 magnet in external magnetic field are studied. ● Influence of the anisotropic biquadratic exchange interaction is examined. ● It is shown that the relative magnetization does not depend on temperature. ● Two branches of the spin excitation spectrum are determined. ● Dependence of critical temperature from anisotropy constants are built. Corresponding author: Peter Sayko.