Magnetic order at a single-crystal surface in the diffuse-scattering theory

Physica B 334 (2003) 21–43

Magnetic order at a single-crystal surface in the diffuse-scattering theory I. Zasada* ! z ul. Pomorska 149/153, Łod! ! z PL 90-236, Poland Solid State Physics Department, University of Łod! Received 22 July 2002; received in revised form 24 December 2002

Abstract A theoretical description of incoherent spin-dependent multiple scattering of electrons at a magnetically disordered single-crystal surface is reported. A formalism in which the spin operators specify the magnetic state of a surface atom is used for the description of magnetic order at the surface. The theory is based upon the concepts used in multiple scattering spin-dependent diffuse LEED theory (DSPLEED) theory. In the present considerations, this theory is extended to the case of magnetic materials by using the time-independent Dirac equation with an effective magnetic field. Thus, an expression for incoherent spin-dependent intensity for magnetic material is obtained. It depends on the Fourier transform on the surface lattice of the spin-pair correlation function and, as a consequence, on the magnetic properties of the surface. The equations for the description of magnetization and various correlation functions in the frame of effective field theory are derived and the results of the numerical calculations are presented for the particular case of Nið1 0 0Þ surface. The spin–orbit induced and exchange asymmetries are calculated. It is found that the magnetic DSPLEED is sensitive to the properties of the surface characterized by the spin-pair correlation functions. Thus, it is demonstrated that the magnetic DSPLEED can be an effective method in the investigation of critical behaviour of magnetic surfaces. r 2003 Elsevier Science B.V. All rights reserved. PACS: 61.10.Dp; 61.14.Hg; 75.25.+z Keywords: Low energy electron diffraction; Electron–solid interaction; Magnetic order

1. Introduction In the diffuse neutron scattering measurements a special magnetic behaviour in a crystal has been observed [1]. It has been shown in this case that fluctuations in a magnetic moment influence the properties of bulk crystals, in particular, the magnetic system becomes inhomogeneous due to *Tel.: +48-426355692; fax: +48-426790030. E-mail address: [email protected] (I. Zasada).

the rapid growth of the magnetization fluctuations in any region of the system when the critical point is approached. Thus, the magnetic structure reflected by means of the neutron critical scattering above the Curie temperature arises from localized clusters of spins which create the shortrange magnetic order. All magnetic scattering is diffuse above the Curie temperature but, because of spin clustering, this scattering is modulated producing the structured intensities around each Bragg scattering position, including the origin [2].

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(03)00004-8

22

I. Zasada / Physica B 334 (2003) 21–43

The long-range spin order in the magnetic system sets in progressively as the temperature is lowered below the Curie temperature and the magnetic scattering from spins forming long-range order appears as sharp peaks at Bragg positions if the magnetic and the nuclear unit cells are identical. However, the residual diffuse magnetic scattering remains even at room temperature [3], which indicates that the magnetic order is not perfect. The analogue behaviour has to be expected in the surface region. Spin polarized low energy electron diffraction (SPLEED) has established itself as a powerful method for studying geometrical, electronic and especially magnetic properties of solid surfaces. In particular, the critical behaviour of the surface magnetization has been derived from experimental data [4–12] via comparison of measured and exchange-induced scattering asymmetry data taken at fixed finite temperatures with their counterparts calculated by the dynamical diffraction theory. On the other hand, the theory describing the spin-dependent low-energy back-scattering of electrons at a single-crystal surface partially covered with a disordered overlayer of elements has been reported recently [13]. It was demonstrated that diffuse SPLEED is sensitive to the geometrical parameters characterizing the substrate/disordered adsorbate system as well as to the parameters related to the statistical distribution of the occupied chemisorption sites. In the DLEED theory [14] as well as in its spindependent version [13] a simple expression for the incoherent intensity has been obtained. It appears as a quadratic form of the effective scattering amplitudes for all possible occupancies of a chemisorption site. The matrix from which this quadratic form is derived is nothing other than the Fourier transform on the lattice of the site– occupancy pair correlation function for a pair of sites with given occupancies. All information concerning the statistical distribution of elements in the overlayer is embodied in this matrix and the correlation function characterizes entirely the short-range order present in the overlayer. It seems natural to extend this theory to the case of magnetically disordered surfaces. Polarization effects due to the exchange interaction are then

introduced by solving the Dirac equation with an effective potential including the magnetic interaction contribution. Magnetic DSPLEED gives the possibility of information about the layer, temperature- and field-dependent magnetization of disordered surfaces, especially in the region of phase transitions. From the theoretical point of view the present theory describes the electron scattering in the case of its diffuse nature and it can be applied to the surface magnetic order investigations due to the neutron experience. The neutron scattering has been successfully used for the measurements of the spin correlation functions in the case of bulk materials while the spinpolarized electron scattering can be applied for their measurements in the surface region. The present paper is organized as follows. Section 2 contains all basic concepts used to construct the magnetic spin-dependent diffuse LEED theory (magnetic DSPLEED). A model of a magnetic order at the surface is presented in Section 2.1. The general description of the spinpolarized LEED calculations for magnetic materials is given in Section 2.2 while the basic ideas of diffuse SLEED including surface magnetizm are gathered in Section 2.3. Section 3 is devoted to the evaluation of the expression for the spin-dependent scattering amplitude in terms of spin operators associated with a particular magnetic configuration of the surface layer. Section 4 deals with the expression of the spindependent scattering amplitude of the semi-infinite monocrystal in presence of a magnetically disordered surface layer in terms of the spin operators. Section 5 brings the numerically calculable expression for the spin-dependent incoherent intensity of electrons back-scattered at the crystal with the magnetic disorder in the surface layer. Section 6 presents the method of the calculation of correlation functions characterizing the magnetic fluctuations in the surface layer. Section 7 collects the numerical results which allow to discuss the surface magnetization phenomenon in the particular case of Nið1 0 0Þ: Section 8 discusses the perspectives of the presented approach and gives some final remarks.

I. Zasada / Physica B 334 (2003) 21–43

2. Basic concepts 2.1. Model of a magnetic order at the surface In the case of a semi-infinite solid with a perfect two-dimensional periodicity the solution of the scattering problem naturally proceeds in three steps: (i) scattering by a single muffin-tin sphere (crystal atom), (ii) multiple scattering within a monoatomic layer parallel to the surface and (iii) multiple scattering between layers. Thus, the scattering amplitude is associated with a particular configuration of the layers forming the investigated system. In the present case, we consider a perfectly ordered monocrystal but we introduce the fluctuations in the arrangement of magnetic moments in the crystal surface layers. The magnetic configuration of a monoatomic layer is defined by associating with each crystallographic site specified by the translation vector Ti ; a vector jpi S in which pi can take a set of discrete values p ¼ 1; 2; y; 2S: Quantity S represents the spin and it can take the values 12; 1; 2; etc., depending on the considered magnetic material. State jpi S has the following vector representation: 2 3 2 3 0 0 6^7 6^7 6 7 6 7 jpj ¼ 0S ¼ 6 7 ; jpj ¼ 1S ¼ 6 7 ; y; 405 415 1 j 2 3 1 607 6 7 jpj ¼ 2SS ¼ 6 7 : 4^5 0

0

j

ð1Þ

j

A particular configuration of the monoatomic layer is characterized by the state vector: jp1 ; p2 ; y; pN S ¼ jp1 S#jp2 S#?#jpN S

ð2Þ

in which N denotes the number of crystallographic sites in the considered layer. It is assumed that the set of these vectors generates an Euclidean vector space in which a scalar product is defined by /p1 ; p2 ; y; pN jp01 ; p02 ; y; p0N S ¼ dp1 p01 ; dp2 p02 ; y; dpN p0N : Any state jp1 ; p2 ; y; pN S of the system is orthogonal to any other state jp01 ; p02 ; y; p0N S of this system because it is represented by the vector

23

consisting of zeros and a single one whose place defines explicitly the state of the system. Let us consider an electron moving towards a target being the semi-infinite monocrystal consisting of the above-described layers. The scattered part of its wave function depends on the configuration of the target. The evaluation of this configuration requires the knowledge of the Hamiltonian describing the considered system. As an example the Ising Hamiltonian written for the arbitrary chosen spin can be taken: 1X Jij S# zi S# zj ð3Þ H# ¼  2 i;j where Jij is the exchange integral for the nearest neighbours. Assuming that the state vector equation (2) is the eigenstate of the Hamiltonian equation (3) it has to be an eigenstate of the operator S# zi : Thus, # ¼ 0 and the operator S# z is defined we have ½S# zi ; H i as a matrix of the ð2S þ 1ÞN ð2S þ 1ÞN dimension: S# z ¼ 11 #?#1i1 #Sz #1iþ1 #?#1N i

i

where 1i is the unit matrix for site i of the ð2S þ 1Þ ð2S þ 1Þ dimension and the operator Szi has the following matrix representation: 1 0 S y 0 C B S1 C B C B z C: B Si ¼ B & ð4Þ C C B S þ 1 A @ 0 y S i Taking into account the definition equation (1) of state jpi S we have S# zi jpi S ¼ ðS þ pi Þjpi S and from Eqs. (1) and (4) we receive the following relation: S# z jp1 ; p2 ; y; pN S ¼ Sz jp1 ; p2 ; y; pN S i

i

Siz

where is the eigenvalue of the spin on site i ðSiz ¼ S; S  1; y; S þ 1; SÞ and S# zi is the operator of the z component of spin on site i constructed according to Eq. (4). Now taking the above discussion into account we can write 1X H¼ Jij Siz Sjz ; ð5Þ 2 i;j

24

I. Zasada / Physica B 334 (2003) 21–43

The electron–solid interaction Hamiltonian contains a spin-dependent contribution. Thus, the diffracted beams exhibit, in addition to intensity variations, spin-polarization effects, which can manifest themselves into two forms: (1) as an asymmetry in the scattered intensities for polarized incident beams of opposite spin alignment; (2) for unpolarized incident beam, as spin polarization of the diffracted beams [15]. There are two types of spin-dependent interaction mechanisms relevant for LEED: (1) spin–orbit coupling and (2) the exchange interaction between the LEED electron and the ground state electrons, in the case of magnetic structures. The spin effects in the scattering of spinpolarized low-energy electron diffraction by the surface manifest themselves by a reversal of the incident beam polarization as well as by a reversal of the direction of surface magnetization. Thus, one has to distinguish for each diffracted beam four intensities Ims ; where s ¼ ðmkÞ refers to the direction of the incident beam polarization and m ¼ ðmkÞ corresponds to the direction of the effective magnetic field parallel and antiparallel to the surface magnetization axis. These four intensities Ims define three scattering asymmetries:

0; leaving Aso which is the spin–orbit induced ‘‘up/ down’’ asymmetry coming from the fact that spinup s ¼ ðmÞ and spin-down s ¼ ðkÞ electrons experience different effective scattering potentials due to a different sign of the spin–orbit coupling term [13]. For magnetic materials the exchange term reverses its sign changing the direction of the magnetic field. This implies that Ims ¼ Iks so the magnetic exchange effects approximately cancel and still leave the spin–orbit coupling as a dominant origin of Aso : However, Aex and Au are now in general non-zero. If the spin–orbit coupling can be ignored Aso ¼ Au ¼ 0 and Aex reduces to a purely exchange induced asymmetry. As for Au ; it can be non-zero only if both the spin–orbit coupling and the magnetic exchange interaction are present. It is obvious from its definition that Au is an asymmetry obtained from an unpolarized incident beam upon the reversal of the magnetization direction. In the present paper, we focus on the spinpolarized electrons back-scattered at the magnetic surface. Thus, in general the spin–orbit as well as the exchange interaction are present and the scattering problem has to be resolved by the multiple scattering theory which incorporates both interactions. An accurate description of the spin– orbit and the scalar relativistic effects on the elastic scattering requires the four-component Dirac equation [15]. The elastic scattering by the magnetically ordered electrons of the solid can be formulated as an effective one-electron problem involving a complex non-local potential matrix. In a local spin density approximation one arrives at one-electron Dirac equation [16]:

Aso ¼ ðImm þ Imk  Ikm  Ikk Þ=I;

½aðirÞ þ b þ eV ð~ r Þ þ eBð~ r Þscð~ r Þ ¼ Eo cð~ rÞ

Aex ¼ ðImm þ Ikk  Ikm  Imk Þ=I; Au ¼ ðImm þ Ikm  Imk  Ikk Þ=I:

in atomic unites, where Bð~ r Þ is the effective magnetic field, Eo is the total energy of electron, cðrÞ is a four component spinor and e ¼ jej: ak ; b and sk are ð4 4Þ matrices which can be expressed in terms of the ð2 2Þ Pauli matrices: ! ! 0 1 0 i r1 ¼ ; r2 ¼ ; 1 0 i 0 ! 1 0 r3 ¼ 0 1

which is called the Hamiltonian although it does not contain the operators and it will be used in further considerations as a Hamiltonian describing our system. 2.2. Electron elastic scattering at the magnetic surface

ð6Þ

The physical meaning of these asymmetries is indicated by the subscripts: so for spin–orbit, ex for exchange and u for unpolarized. Thus, these quantities correspond to the different interactions which take place during the spin-dependent elastic scattering process. For non-magnetic materials, the only relevant spin-dependent mechanism is spin–orbit coupling, and consequently Aex ¼ Au ¼

ð7Þ

I. Zasada / Physica B 334 (2003) 21–43

25

b ¼ r3 #I;

ak ¼ r1 #rk ;

jp1 ; y; pN S are, respectively, given by Z Z % 1 ; y; pN ;~ Aðp1 ; y; pN Þ ¼ d~ r d~ r 0 cðp r ÞAð~ r ;~ r0Þ

sk ¼ I#rk ;

k ¼ 1; 2; 3

r 0 Þ;

cðp1 ; y; pN ;~

as the tensor products:

I ð10 01Þ is the unity matrix. The effective electrostatic potential V ð~ r Þ incorporates the Coulomb interaction with the nuclei and the electrons of the solid, exchange and correlation effects. The spin–orbit coupling is already inherent in the Dirac equation. The effective magnetic field Bð~ r Þ arises from the exchange and correlation effects in the case of ferromagnetic ordering. It is generally assumed that the muffin-tin shape is taken for V and B: Thus, the Dirac Hamiltonian equation (7) has spherically symmetric V and B but it is itself not spherically symmetric. The usual relativistic partial wave analysis [17] is therefore not applicable. The suitable generalization was given in Refs. [15,17] and is used in this paper. 2.3. Spin-dependent diffuse low-energy electron diffraction at the magnetic surface On the basis of the two previous sections, the construction of the diffuse LEED theory including spin–orbit and exchange interaction is now presented. Theoretical description of spin-dependent incoherent scattering of electrons at a single-crystal surface with the magnetic fluctuations is the main goal of this presentation. Let us consider an electron of wave vector k~; moving towards a target. Its wave function Ck (more especially the scattered part of this wave function) depends on the configuration jp1 ; p2 ; y; pN S of the target. So the measurement of a physical quantity A ascribed to the electron is the statistical average [13]: /AS ¼

2S X p1 ¼0

?

2S X

lðp1 ; y; pN ÞAðp1 ; y; pN Þ

pN ¼0

in which the quantum average of A and the probability l of finding the configuration state

z lðp1 ; y; pN Þ ¼ Tr½rS1z ySN :

In these relations Að~ r ;~ r 0 Þ is the ~ r —representation of the operator corresponding to the physical quantity A and r is the density matrix of the system. Using two last expressions /AS can be rewritten as Z Z % r ;~ /AS ¼ d~ r d~ r 0 Tr½rWð~ r 0 ÞAð~ r ;~ r 0 ÞWð~ r 0 Þ; ð8Þ where the field-like operator W acting on states jp1 ; p2 ; y; pN S is given by X z Wð~ rÞ ¼ S1z ?SN cðp1 ; y; pN ;~ r Þ: ð9Þ p1 ?pN

In order to separate incoherent scattering from the diffraction Wð~ r Þ is decomposed into two parts: Wð~ r Þ ¼ /Wð~ r ÞS þ dWð~ rÞ

ð10Þ

/Wð~ r ÞS denotes the statistical average of the wave functions cðp1 ?pN ;~ r Þ and dWð~ r Þ takes into account the fluctuations associated with the difference between the average magnetic configuration and the magnetic configurations actually ‘‘seen’’ by incident electrons. It follows that /dWð~ r ÞS ¼ 0: It can be easily shown that /AS can be decomposed into /AS ¼ /AScoh þ /ASincoh

ð11Þ

in which the coherent and incoherent parts of the average value of A are, respectively, given by Z Z % r ÞSAð~ /AScoh ¼ d~ r d~ r 0 /Wð~ r ;~ r 0 Þ/Wð~ r 0 ÞS; /ASincoh ¼

Z

Z d~ r

% r ÞAð~ d~ r 0 Tr½rdWð~ r ;~ r 0 ÞdWð~ r 0 Þ:

The above construction uses the ideas applied in the case of the DLEED and DSPLEED theory including multiple scattering processes and based on the formalism in which occupation operators s specify the occupancy of a chemisorption site [13,18]. Here this operator is replaced by S z which specifies the magnetic state of the atoms forming

I. Zasada / Physica B 334 (2003) 21–43

26

the system. In order to take into account the spin effects during the scattering process involving the magnetic surface, the Dirac equation (7) will be . used instead of the Shrodinger equation [14] and Dirac equation [13].

(2) the second term is more complicated and it takes into account the multiple scattering processes. These processes take place when the scattering of an electron at site i is the last elementary process which follows the effective scattering at other sites.

3. Spin-dependent scattering amplitude of the layer with a magnetic disorder

Let us call k~ni the wave vector of the electron incident on the magnetic layer. The superscript n is þ or  according to whether the z component of the wave vector perpendicular to the surface is positive or negative. It is assumed that the incident electron is described by a Dirac plane wave of the form umk expðik~ni~ r Þ: Including incident and scattered waves the total wave function in terms of field-like operator Wð~ r Þ introduced in Section 2.3 is given by

The relativistic scattering amplitude of the magnetic layer with a given arrangement of magnetic moments will be now evaluated. All the details of such calculations will not be presented here because they are based on the same scheme as in the case of DSPLEED theory [13]. Thus, only the most crucial points of this evaluation will be presented and all the differences coming from the facet that we use here the Dirac equation (7) will be discussed in detail. The relativistic transition matrix Tðp1 ; p2 ypN Þ of the layer for the magnetic configuration state jp1 ?pN S can be expressed in terms of the effective transition matrices tni ðp1 ?pN Þ: Tðp1 ?pN Þ ¼

N X

tni ðp1 ?pN Þ:

ð12Þ

i¼1

These transition matrices provide the wave scattered at a site i in the presence of the other sites and each of them is given by [13,18] X tni ðp1 ?pN Þ ¼ ti ðpi Þ þ ti ðpi ÞGo ðEÞ tnq ð13Þ qai

where Go ðEÞ is the Green operator of the Dirac free electron and E is the energy of the incident electron. Relation equation (13) shows that the effective transition matrix tni ðp1 ?pN Þ is the sum of two terms: (1) the first term ti ðpi Þ is the relativistic transition matrix of the isolated site i in the magnetic state pi (this term is associated with a single scattering process at this site). This transition matrix is obtained by solving the Dirac equation (7). The suitable procedure to do this was given in Refs. [15,17],

~n r Þ Wð~ r Þmk mk ¼ umk expðik i ~ Z X ~n0 ~n dkf jj Mmk þ mk ðk f ’k i Þ n0

0

expðik~ni ~ r ÞYðn0 ðz  R> ÞÞ

in which the matrix elements of

Mmk mk

ð14Þ are given by

ip 0 ~n0 ~n ~Þ expðik~ni R Mmk mk ðk f ’k i Þ ¼ kf > N X 0 ~Þ ð15Þ Tnq ðk~nf ’k~ni Þ expðik~ni R

q¼1

Wmk mk ;

Mmk mk

n

and Tq are operators acting in the magnetic configuration space. The latter operator is given by 0 Tnq ðk~nf ’k~ni Þ X 0 z n ¼ S1z ?SN tq ðp1 ?pN ; k~nf ’k~ni Þ:

ð16Þ

p1 ?pN mk In subsequent developments, /WSmk mk and dWmk will be expressed in terms of the spin-dependent effective scattering amplitudes of sites and of the factor characterizing the distribution of magnetic moments in the considered layer. This requires the mk preliminary evaluation of /Mmk mk S and dMmk and, n consequently, the evaluation of /Tq S and dTnq : It is not possible to obtain the exact expressions for these quantities. However, /Tnq S and dTnq can be determined by using the well-known approximations presented in Refs. [13,18].

I. Zasada / Physica B 334 (2003) 21–43

Using the calculation scheme presented in Refs. [13,18] a well-known expression for /Mmk mk S is obtained, i.e.: ~n ~n /Mmk mk ðk f ’k i ÞS X 8ip2 X 0 ~Þ ¼ expðik~nig R Cðl 12 j; m812; 712Þ Akg> fgg kmk0 m0 0

0

Yl;m81=2 ðk~nig Þ½ð1  X Þ1 /T1 Skmk0 m0

n

Cðl 12j; m  s; sÞYl;ms ðk~ni Þ ~Þdðk~f jj  k~ijj  gÞ;

expðik~n R i

1 s¼72



Cðl 0 12 j 0 ; m0

 s; sÞ

X m

ð18Þ

where the lattice summation remains the same as in the spinless case while the dimension of X is doubled to 2ðlmax þ 1Þ2 2ðlmax þ 1Þ2 ; where lmax ¼ supðlÞ is taken into account. The Green function is determined due to the definition given by Pendry [19]: X 0 00 0 00 ~R ~m Þ ¼ Glml 00 m00 ðR 4pð1Þ1=2ðll l Þ ð1Þm þm l 0 m0

Bl ðl 0 m0 ; l 00 m00 ÞDl 0 m0 ;

p¼0

/S1z St1 ðpÞ;

0

ip 0 ~Þ expðik~nf R kf > 2S X 0

F ðp; k~nf ’k~ni Þ p¼1 0 ~Þ ð20Þ

I1 ðp; k~nf ’k~ni Þ expðik~ni R

dMmk mk is expressed in terms of a product of two factors: 0 (1) the factor I1 ðp; k~nf ’k~ni Þ characterizes the spin-dependent scattering properties of the scatterer in magnetic state p in the presence of the other scatterers and it is given on the basis of free spherical waves as follows: I1 ðp; k~nf ’k~ni Þ 2 X n ~n0 Cðl 12 j; m  s0 ; s0 ÞYl;ms ¼ 0 ðk f Þ p kmk0 m0

½t1 ðpÞð1  ZÞ1 kmk0 m0

Cðl 12 j; m  s; sÞYl;ms ðk~ni Þ

ð21Þ

where the matrix elements of Z are given by X Zkmk0 m0 ¼ Cðl 12 j; m812; 712Þ s¼71=2

Cðl 0 12 j 0 ; m0  s; sÞ X ~m Þ

expðik~ijj R m

where Bl ðl 0 m0 ; l 00 m00 Þ are the Gaunt coefficients and Dl 0 m0 are the structure factors. The evaluation of the Green function operators is based on the Kambe’s method used in LEED theory [19], magnetic SPLEED theory [16] as well as in DLEED theory [14] and DSPLEED theory [13]. The average value /T1 S can be rewritten in the form 2S X

~n ~n dMmk mk ðk f ’k i Þ ¼

0

~m Þ expðik~ijj R

~R ~m Þ;

Gl;m81=2;l 0 ;m0 s ðR

/T1 S ¼

dMmk mk denotes the scattering amplitude deviation connected with the magnetic fluctuations in the considered layer. Using again the calculation scheme presented in Refs. [13,18] an expression for dMmk mk in the momentum representation is obtained, i.e.:

ð17Þ

where s ¼ ðmkÞ while the summation over k corresponds to the summation over two configurations: k ¼ l  1 for j ¼ l þ 1=2 and k ¼ l for j ¼ l  1=2 (cf. the definition of the Clebsh– Gordan coefficients [17]). The elements of the multiple scattering matrix X are given by X Xkmk0 m0 ¼ /T1 S Cðl 12 j; m812; 712Þ

27

ð19Þ

~R ~m Þ/Tn S;

Gl;m81=2;l 0 ;m0 s0 ðR 1

ð22Þ

0 F ðp; k~nf ’k~ni Þ

characterizes the (2) the factor distribution of magnetic moments in the layer: 0 F ðp; k~nf ’k~ni Þ ¼

N X

½Siz  /Siz S

i¼1 0

~i  T ~1 Þ: ð23Þ

exp ½iðk~nf ’k~ni ÞðT The set of operators Siz can be regarded as a discrete field which defines the magnetic composition of the 2D lattice sites. Then, Siz  /Siz S is the deviation of this field from its average and

I. Zasada / Physica B 334 (2003) 21–43

28

0 F ðp; k~nf ’k~ni Þ is the Fourier transform on the lattice of this deviation.

4. Spin-dependent scattering amplitude of the mono-crystal with a magnetic disorder at the surface The scattering amplitude of a single-crystal surface covered with a layer which is different form the bulk is given by the well-known relation: A

mk

crystal can be found in an elaboration on the classical SPLEED [20] and it will not be discussed in this paper. The spin-dependent scattering amplitude equation (24) is associated with a particular magnetic configuration of the system. It has to be expressed now in terms of spin operators Siz : X z ~ ~þ Amk S1z ?SN mk ðk f ’k i Þ ¼ p1 ;y;pN

~þ

Amk ðp1 ; y; pN ; k~ f ’k i Þ

~þ ðp1 ; y; pN ; k~ f ’k i Þ

and using again the technique from the DLEED [14] and DSPLEED [13] theory one arrives at two relations for the spin-dependent scattering amplitude of the magnetic crystal in the presence of surface magnetic fluctuations. The first one is connected with the coherent scattering processes in the system:

~þ ¼ M mk ðp1 ; y; pN ; k~ f ’k i Þ ~ þ ½1 þ M mk ðp1 ; y; pN ; k~ f ’k i Þ ~ ~þ

½1  Mc mk mk ðk f ’k i Þ ~ 1

M mk ðp1 ; y; pN ; k~þ f ’k i Þ ~ ~þ

Mc mk mk ðk f ’k i Þ ~þ

½1 þ M mk ðp1 ; y; pN ; k~þ f ’k i Þ:

ð25Þ

ð24Þ

0 The scattering amplitude M mk ðp1 ; y; pN ; k~nf ’k~ni Þ is connected with a surface layer with different scattering properties with respect to the perfect crystal on top of which this layer is placed. The difference in the surface layer properties can be caused by various reasons as, for example, interlayer spacing different from the bulk one, different chemical composition, imperfect or disordered arrangement of elements, different magnetization-like in the present considerations, etc. Of course, there can be more than one layer which exhibits behaviour untypical of the bulk. In the case of the surface magnetic properties, it was reported (Ref. [10] and references therein) that for the Feð1 1 0Þ surface at least two such layers have to be taken into account. However, in the present considerations only one layer is assumed to have a different magnetic moment characterized by the fluctuations. It seems at first to be enough for describing the Ni surface [9] which serves as an example here. Obviously, in further developments more layers can be considered by ‘‘putting’’ the additional layer on the top of the construction defined by relation (24). The expression for the spin-dependent scattering matrix Mc mk mk of a magnetic semi-infinite single

~ ~þ /Amk mk ðk f ’k i ÞS mk ~ ~ ~þ ~ ¼ /Mmk mk ðk f ’k i ÞS þ ½1 þ /Mmk ðk f ’k i ÞS

~ ~þ

½1  Mc mk mk ðk f ’k i Þ ~þ ~ 1

/Mmk mk ðk f ’k i ÞS ~ ~þ

Mc mk mk ðk f ’k i Þ ~þ ~þ

½1 þ /Mmk mk ðk f ’k i ÞS:

ð26Þ

The second one describes the incoherent scattering processes in the system caused by the presence of the magnetically disordered surface layer: ~ ~þ dAmk mk ðk f ’k i Þ mk ~ ~ ~þ ~ ¼ dMmk mk ðk f ’k i Þ þ dMmk ðk f ’k i ÞF2

~þ ~þ þ F1 dMmk mk ðk f ’k i Þ ~þ ~ þ F1 dMmk mk ðk f ’k i ÞF2 ; where mk ~ ~ F1 mk mk ¼ ð1 þ /Mmk ðk f ’k i ÞSÞ

~ ~þ

½1  Mc mk mk ðk f ’k i Þ ~þ ~ 1

/Mmk mk ðk f ’k i ÞS ~ ~þ

Mc mk mk ðk f ’k i Þ

ð27Þ

I. Zasada / Physica B 334 (2003) 21–43 mk ~ ~þ F2 mk mk ¼ ½1  Mc mk ðk f ’k i Þ

~þ ~ 1

/Mmk mk ðk f ’k i ÞS ~ ~þ

Mc mk mk ðk f ’k i Þ ~þ ~þ

ð1 þ /Mmk mk ðk f ’k i ÞSÞ: The four terms in expression (27) can be interpreted in the same way as in the case of general DLEED [14] and DSPLEED [13]. The first term is associated with a single backward incoherent scattering event at the surface layer. The second term takes into account processes in which a single forward incoherent event from the bulk through the surface layer to the vacuum takes place after multiple coherent reflections between the surface layer and the bulk. The third term is associated with processes in which a single forward incoherent scattering event from the vacuum through the surface layer to the bulk takes place before multiple coherent reflections between the surface layer and the bulk. The fourth term describes processes in which a single backward incoherent scattering occurs between multiple coherent reflections between the surface layer and the bulk. Now, following the evaluation of the general DLEED [14] and DSPLEED [13] theory we can calculate the spin-dependent coherent and incoherent wave defined in Eq. (10) and the decomposition equation (11) can be applied to the particular case where A is the electron probability current density at ~ r o : The coherent part of the current is determined by the statistical averages of the wave function while the incoherent part is described by the fluctuations and it corresponds to the magnetic diffuse SPLEED which is a subject of the present considerations.

29

The summation over ~ g is performed over all the reciprocal lattice vectors for which the component kig> in the vacuum is real. Each term of this sum is the spin-dependent relative intensity of a diffracted beam labelled by ~ g: The calculation of incoherent intensity follows the same scheme as in the case of the general DLEED theory [4] and leads to the expression of the spin-dependent relative intensity per unit of solid angle in the direction of the detector in the form: ~ ~þ Iincoh mk mk ðk f ’k i Þ  2 k2 4p ~ ’k~þ Þj2 S fg> : ð29Þ ¼ lim ð k ki /jdAmk i mk fg s-N s ki> To obtain the expression useful for the numerical calculations, the quantity dAmk mk in Eq. (29) should be replaced by its relation (27), i.e. Iincoh mk ðk~ ’k~þ Þ mk

f

i

2S X 2S 4p ki X ~þ ¼ Gðp; p0 ; k~ f ’k i Þ Aki> p¼0 p0 ¼0 4

~ ~þ

K% 1 mk mk ðp; k f ’k i Þ 0 ~ ~þ

K1 mk mk ðp ; k f ’k i Þ;

where ~þ Gðp; p0 ; k~ f ’k i Þ N X

¼

½/S1z Siz S  /S1z S/Siz S

i¼1

~þ ~ ~

exp½iðk~ f  k i ÞðT i  T 1 Þ

The flux of coherent back-scattered electrons per unit of surface and per unit of incident flux can be calculated. This calculation yields X kig> ~ ~þ 2 Icoh mk j/Amk ð28Þ mk ¼ mk ðk ig ’k i ÞSj : k i> fgg

ð31Þ

describes the distribution of the magnetic moments in the surface layer, and K1 mk mk ~þ ¼ I1 ðp; k~ f ’k i Þ þ

5. Coherent and incoherent spin-dependent back-scattered intensities

ð30Þ

X

~ I1 ðp; k~ f ’k ig1 Þ

fg1 g



~ ~þ F2 mk mk ðp; k ig1 ’k i Þ

þ

X kf > ~ ~þ ~þ ~þ F1 mk mk ðk f ’k fg2 ÞI1 ðp; k fg2 ’k i Þ k fg g fg> 2

X X kf > ~ ~þ F1 mk þ mk ðk f ’k fg2 Þ k fg > 2 fg g fg g 1

2

mk ~ ~ ~þ

I1 ðp; k~þ fg2 ’k ig1 ÞF2 mk ðk ig1 ’k i Þ

ð32Þ

30

I. Zasada / Physica B 334 (2003) 21–43

describes the incoherent scattering at the reference site in the presence of other surface scatterers (multiple scattering in the surface layer) and of the bulk (multiple scattering between the surface layer and the bulk). The last relation can be interpreted in the same way as Eq. (27) under the condition that the surface layer is replaced with a single surface layer scatterer located at the reference site 1. A in expression Eq. (30) is the area of the unit cell of the crystal surface lattice and N is the number of all the considered surface layer sites. Relation equation (30) is equivalent to those obtained for the incoherent scattering intensities of electrons back-scattered at a single-crystal surface partially covered with a disordered layer both in the case of DLEED theory [14] and DSPLEED theory [13]. In order to underline clearly the differences arising from the introduction of the magnetic disorder instead of the geometrical one considered in the above-mentioned theories all stages of the spin-dependent magnetic DLEED calculations will be now pointed out and analysed. First, the knowledge of the relativistic transition matrices in angular momentum representation for the reference site in any magnetic state is required. These atomic t-matrices have the elements tkmk0 m0 which arise from the solution of the Dirac equation (7) for a single muffin-tin sphere (‘‘crystal atom’’). The approach proposed in Ref. [16] leads to the system of coupled differential equations for the radial functions which for B ¼ 0 are seen to reduce to the standard central field Dirac equations [17]. The magnetic field introduces couplings, which depends on m via the matrix elements of r3 (Section 2), corresponding to the chosen magnetic axis. For B normal to the layer t-matrices are diagonal in m but they have off-diagonal elements with respect to k corresponding to the dominant coupling between j ¼ l þ 1=2 and j ¼ l  1=2; and the weak Dl ¼ 72 coupling. For non-magnetic systems, for which these couplings are absent, the elements of t-matrices are ð1=2Þðexpð2idki Þ  1Þ where the quantity dki represents the relativistic scattering phase shift ðk ¼ l  1 for j ¼ l þ 1=2 and k ¼ l for j ¼ l  1=2Þ [13]. In the case of a magnetic field with an arbitrary direction the tmatrices are no longer diagonal in m: They can be obtained from the m-diagonal matrix by applying

the appropriate rotation representation in the orbital angular momentum and the spin space [21]. The next step is connected with the calculations ~þ of the relativistic transition matrix I1 ðp; k~ f ’k i Þ in the momentum representation equation (21). The elements of this matrix are expressed by means of the transition matrices discussed above, Clebsh–Gordan coefficients defined in the usual way [17] and the Green functions due to Pendry’s notation [19]. Using now expression (32) we can determine the matrix regarded as the effective transition matrix of the surface layer scatterer with a given magnetic moment p located at the reference site in the presence of the other surface layer scatterers and the rest of the semi-infinite crystal. Of course, its calculations require the mk knowledge of matrices F1 mk mk and F2 mk whose determination is related to the classical SPLEED techniques [20]. Thus, according to Eqs. (30) and (31) it remains to perform the Fourier transforms on the surface lattice of the spin-pair correlation functions Gðp; p0 Þ which are the magnetic structure factors of the surface layer. This point is original with respect to the diffuse scattering theories [13,14] and it will be developed in the next section.

6. Magnetization and the correlation functions of the surface layer One of the most convenient techniques which has been used for studying the magnetic properties of the surfaces is the effective field theory (EFT) with auto-correlations [22]. The effective field theories in magnetism were developed in the 1950s and early 1960s and are still often used. The first attempt to unify the various EFT was given in the book by Smart [23]. Later, new techniques were developed, namely, the differential and the integral operator methods, which have been effectively applied to the calculations based on several exact identities for the Ising model [24–26]. The EFT method has been adopted for the surfaces of a semi-infinite bulk and for thin films with different surface conditions taken into account [22,27–29]. The magnetic properties of the single-crystal surface layer are studied here in the frame of the integral operator technique [26].

I. Zasada / Physica B 334 (2003) 21–43

31

6.1. Theory

site i; then the Hamiltonian can be decomposed as follows:

Let us distinguish a group of spins (cluster) in the system which may contain a limited or an unlimited number of spins. Then, the Hamiltonian describing the considered system can be decomposed into a sum of two terms [26]:

H ¼ H0 þ H 0 ¼ h0 Siz þ H 0 ;

ð36Þ

where X h0 ¼ Jij Sjz :

ð37Þ

*

*

H0 which describes the interaction of the spins inside the cluster and the interaction of the cluster with its environment; H 0 which relates to the part of the system not containing the spins from the cluster.

Let us denote by f0 an arbitrary function of the spin variable within the cluster, defined on all spins belonging to it and by ff g an arbitrary function of spin variables outside the cluster, defined on an arbitrary number of spins. Then, the multi-spin correlation function is defined as 1 /ff gf0 S ¼ Tr½ff gf0 expðbðH0 þ H 0 ÞÞ; ð33Þ Z where b ¼ 1=kT and Z is the partition function. The trace in the above equation can be calculated in two steps: first, the trace is taken on cluster states only ðTr0 Þ and next, the trace on the states from the environment is calculated ðTr0 Þ . Thus, Eq. (33) can be rewritten in the form 1 /ff gf0 S ¼ Tr0 ½ff g Tr0 ð f0 expðbH0 ÞÞ Z ð34Þ

expðbH 0 Þ and, after some simple transformations, we have [26]   Tr0 ð f0 expðbH0 ÞÞ /ff gf0 S ¼ ff g ; ð35Þ Tr0 ðexpðbH0 ÞÞ which is a more convenient form of Eq. (33) because the local trace on the cluster ðTr0 Þ can often be calculated exactly, while the calculation of the external mean value needs some approximations in the majority of cases. Let us consider now a monoatomic magnetic layer with spin S ¼ 1=2 described by the Hamiltonian equation (5), where Siz ¼ 71=2 are the spin variables in the ith 2D lattice site. If the cluster contains only one spin on

jAi

If we choose f0 function in Eq. (35) as Siz ; we can write the following equation: * + P Tri ðSiz expðbSiz jAi Jij Sjz Þ z P /ff gSi S ¼ ff g : Tri ðexpðbSiz jAi Jij Sjz Þ ð38Þ From the above equation, after trace calculations, we get * !+ 1 1 X z z b /ff gSi S ¼ ff g tanh Jij Sj ; ð39Þ 2 2 jAi which is known in the literature as Callen’s identity [30,31]. Now, putting ff g ¼ 1 in Eq. (39) we receive the equation for the magnetization: * !+ 1 1 X z z tanh b Jij Sj /Si S ¼ ð40Þ 2 2 jAi and putting ff g ¼ Siz we receive the equation for the correlation between two spins located at the lattice sites i and j: * !+ 1 X z z z1 z /Sj Si S ¼ Sj tanh b Jij Sj : ð41Þ 2 2 jAi For further considerations we use the integral operator method [26]. Then, the statistical average of any function F ðxÞ can be expressed by an integral, using the Dirac d distribution function: * !+ Z N X z /F ðxÞS ¼ dxF ðxÞ d x  Jij Sj : ð42Þ N

jAi

Taking into account the above general relation, Eq. (40) can be rewritten in the following form:   Z 1 N 1 dx tanh bx /Sjz S ¼ 2 N 2 * !+ X z

d x Jij Sj ð43Þ jAi

I. Zasada / Physica B 334 (2003) 21–43

32

P and by expressing dðx  jAi Jij Sj Þ as the Fourier transform of unity in the sense of distribution (the so-called integral representation of the Dirac dfunction), we have   Z N Z 1 N 1 1 /Siz S ¼ dx tanh bx dt 2 N 2 2p N * !+ X z Jij Sj

expðixtÞ exp it : ð44Þ jAi

The evaluation of /expðit out by noting that * !+ X z exp it Jij Sj * ¼

jAi z Y

P

jAi

;

ð45Þ

jAi

where z is the number of nearest neighbours in the considered system. For further calculations, first, the thermodynamic average equation (45) has to be found. Using the exact identity [30] exp ðitJij Sjz Þ ¼ cosððt=2ÞJij Þ  2iSj sinððt=2ÞJij Þ we can write * !+ X z Jij Sj exp it * ¼

jAi

z Y jAi

/Siz S mb   Z N Z 1 N 1 1 ¼ dx tanh bx dt expðixtÞ 2 N 2 2p N h   t  i8 t 

cos Jb  2iSjz sin Jb 2 2 b h t   t  i4 

cos Jb  2iSjz sin Jb ð jAiÞ; 2 2 s ð47Þ

Jij Sjz ÞS is carried

+

expðitJij Sjz Þ

layer, namely,

+ h t   t i z cos Jij  2iSj sin Jij 2 2

ð46Þ

and this form will be used in all further evaluations leading to the correlation functions. The operator technique, presented here for S ¼ 1=2; can be applied for larger spin values as well [32]. It is also worth noting that this operator method has been applied for multi-atom clusters [33]. 6.2. Application to the case of Ni(1 0 0) surface In the particular case of the semi-infinite Ni single crystal with ð1 0 0Þ surface two equations for the magnetization defined in Eq. (40) have to be constructed; one for the atom i inside the crystal and the other one for the atom i from the surface

/Skz S ms   Z N Z 1 N 1 1 ¼ dx tanh bx dt expðixtÞ 2 N 2 2p N h   t  i4 t 

cos Js  2iSlz sin Js 2 2 s h t   t  i4 

cos Jb  2iSlz sin Jb ðlAkÞ; 2 2 b ð48Þ where Jb is the exchange integral inside the crystal and Js is the exchange integral in the surface layer while /Siz S is denoted by mb for the magnetization inside the crystal and by ms for the magnetization of the surface layer. The powers in Eqs. (47) and (48) are connected with the numbers of nearest neighbours appearing in the considered system Nið1 0 0Þ (see Fig. 1) for the atom in the bulk ðz ¼ 12Þ and in the surface layer ðz ¼ 8Þ: The exchange integral between the surface layer and the next one is taken to be Jb : Making now the assumption that we are taking into account the correlation effects in the surface layer only while we are decoupling the correlations inside the bulk and between the surface layer atoms and the bulk atoms, after standard calculations within integral operator technique [29], we receive the following coupled equations for the bulk and surface magnetization: mb ¼ ð8mb þ 4ms Þb1 þ ½56m3b þ 112m2b ms þ 8mb ð4/S1z S2z S þ 2/S1z S3z SÞ þ 4/S1z S2z S3z Sb2 þ ½56m5b þ 280m4b ms þ 56m3b ð4/S1z S2z S þ 2/S1z S3z SÞ

I. Zasada / Physica B 334 (2003) 21–43

Fig. 1. The structural model of the Ni single crystal with ð1 0 0Þ surface termination.

33

Eqs. (49) and (50) can only be solved if the 2-, 3-, and 4-site correlation functions which are by themselves the solutions of other equations generated by Eq. (41) are known. In fact, Eqs. (49) and (50) are the first ones of an infinite chain of such equations. In order to simplify this set of coupled equations we are going to use a decoupling scheme for the many-site correlation functions. One of the best decoupling techniques is the Matsudaira method [34] based on the cumulant expansions, which enables the spin-pair correlation function calculations to be carried out. The Matsudaira method has been successfully used for thin diluted films [28] and amorphous systems where the fluctuations of the exchange integrals [35] have been taken into account. In the cumulant technique the averages from Eqs. (49) and (50) can be written in the form

þ 112m2b /S1z S2z S3z S þ 8mb /S1z S2z S3z S4z Sb3

/S1z S ¼ /S1z Sc ;

þ ½8m7b þ 112m6b ms

/S1z S2z S ¼ /S1z S2z Sc þ /S1z Sc /S2z Sc ;

þ 56m5b ð4/S1z S2z S þ 2/S1z S3z SÞ

/S1z S2z S3z S ¼ /S1z S2z S3z Sc þ /S1z S2z Sc /S3z Sc

þ þ þ þ þ

280m4b /S1z S2z S3z S þ 56m3b /S1z S2z S3z S4z Sb4 ½4m8b ms þ 8m7b ð4/S1z S2z S þ /S1z S3z SÞ 112m6b /S1z S2z S3z S þ 56m5b /S1z S2z S3z S4z Sb5 ½4m8b /S1z S2z S3z S ð49Þ 8m7b /S1z S2z S3z S4z Sb6 ;

ms ¼ 4mb a1 þ 4ms a2 þ 4/S1z S2z S3z Sa3 þ 4mb ð4/S1z S2z S þ 2/S1z S3z SÞa4 þ 24m2b ms a5 þ 4m3b a6 þ 4mb /S1z S2z S3z S4z Sa7 þ 24m2b /S1z S2z S3z Sa8 þ 4m3b ð4/S1z S2z S þ 2/S1z S3z SÞa9 þ 4m4b ms a10 þ 4m3b /S1z S2z S3z S4z Sa11 þ 4m4b /S1z S2z S3z Sa12 :

ð50Þ

In the above equations the spin variables are numbered in accordance with Fig. 2. The temperature-dependent coefficients bl ðl ¼ 1; 2; y; 6Þ and ak ðk ¼ 1; 2; y; 12Þ have quite complicated forms containing the Jb and Js exchange parameters. Their explicit forms are listed in Appendix A.

þ /S1z S3z Sc /S2z Sc þ /S2z S3z Sc /S1z Sc þ /S1z Sc /S1z Sc S/S1z Sc ^

ð51Þ

in which /S1z Sc ; /S1z S2z Sc ; /S1z S2z S3z Sc ; etc. are the cumulants of order 1, 2, 3, etc., respectively. As a matter of fact, these relations are the definitions of cumulants. Here, we only provide the definitions of cumulants up to the third order; however, the definitions for cumulants of higher order can be easily guessed. In particular, the first-order cumulant is the average magnetization of any site while the second-order cumulant is nothing else than the spin-pair correlation function Gðp; p0 Þ from Eq. (31). Hence, Eqs. (49) and (50) can be expressed by the cumulants of the order 1, 2, 3, and 4. In the generally accepted approximation, introduced by Matsudaira [34], we will neglect the cumulants of order greater than two. In this approximation we have, for instance, the following decoupling of three-spin correlation function: /S1z S2z S3z S Dms ð/S1z S2z S þ /S1z S3z S þ /S2z S3z SÞ  2m3s :

ð52Þ

I. Zasada / Physica B 334 (2003) 21–43

34

C1 C6 C2 8

3

11

2

5

7

10

0

1

6

9

C7 C3 12

C4 C8

4

C5

(a)

a

C9

(b)

Fig. 2. Nine correlation zones considered at the ð1 0 0Þ surface of Ni (a) and the positions of spin pairs in the considered correlation functions Ck (b). Index k ¼ 1; 2; y; 9 corresponds to the correlation zones which are given by the spin-pair distances.

Analogously, all higher-order correlation functions can be decoupled into the two-site correlation function and the magnetization. Now, we will establish a general equation for the two-site correlation function. Starting from Eq. (41) and using the procedure analogous to the one leading to Eqs. (49) and (50) for the bulk and the surface magnetizations, we obtained the following results for the correlation function /S0z Sxz S between the reference site 0 and the arbitrarily chosen site x:

/S0z Sxz SD4mb ms a1 þ ½/S1z Sxz S þ /S2z Sxz S þ /S3z Sxz Sa2 þ ½/S1z S2z S3z Sxz S þ /S2z S3z S4z Sxz S þ /S3z S4z S1z Sxz S þ /S4z S1z S2z Sxz Sa3 þ 4mb ½/S1z S2z Sxz S þ /S2z S3z Sxz S þ /S3z S4z Sxz S þ /S4z S1z Sxz S þ /S1z S3z Sxz S þ /S2z S4z Sxz Sa4 þ 6m2b ½/S1z S2z S þ /S2z Sxz S þ /S3z Sxz Sa5 þ 4m3b ms a6 þ 4mb /S1z S2z S3z S4z Sxz Sa7 þ 4m2b ½/S1z S2z S3z Sxz S þ /S2z S3z S4z Sxz S þ /S3z S4z S1z Sxz S þ /S4z S1z S2z Sxz Sa8 þ 4m3b ½/S1z S2z Sxz S þ /S2z S3z Sxz S þ /S3z S4z Sxz S þ /S4z S1z Sxz S þ /S1z S3z Sxz S þ /S2z S4z Sxz Sa9 þ m4b ½/S1z S2z S þ /S2z Sxz S þ /S3z Sxz S

/S4z Sxz Sa10 þ 4m3b /S1z S2z S3z S4z Sxz Sa11 þ m4b ½/S1z S2z S3z Sxz S þ /S2z S3z S4z Sxz S þ /S3z S4z S1z Sxz S þ /S4z S1z S2z Sxz Sa12 :

ð53Þ

Now, we have to express the correlation functions appearing in the above relation in terms of cumulants and, according to the approximation previously mentioned, we keep only the cumulants of the first and the second order. Then, after some rather laborious calculations, we obtained an equation for the arbitrary spin-pair correlation function containing only magnetization and pair correlations. It is out of the question to give here the explicit form of this equation because of its complicated form. The main thing to be noticed is that, similarly to Eqs. (49) and (50), these equations are algebraic and non-linear with respect to the correlation functions and they can be solved only numerically. 6.3. Numerical results and discussion In Section 6.2, we derived the infinite set of algebraic equations involving the equations for the bulk and the surface magnetization (Eqs. (49) and (50)) and for the various correlation functions of spin pairs with different spin distances (Eq. (53)). In order to make this set of equations solvable we

I. Zasada / Physica B 334 (2003) 21–43

1. l ¼ 1 ðJb ¼ Js Þ means that the magnetic interaction between the neighbouring atoms is homogenous in the whole considered system. Then, Fig. 3a shows the temperature dependence of the magnetization for the surface layer and for the bulk. It can be seen that the surface magnetization is lowered with respect to the

0.5 0.4 m

λ=1 0.3 0.2 0.1 0.0 0

1

2 kT Js

(a)

0.5 0.4 λ = 0.5 m

restrict it to the finite number of distancedependent spin pairs taken into account. Let us introduce the ‘‘zones’’ of correlation connected with the nearest, next-nearest, third-nearest, etc., neighbours of the considered spin in Nið1 0 0Þ system (Fig. 2a). We will consider here nine correlation zones and in this case the set of equations to be solved consists of 11 coupled equations: two for the bulk and the surface magnetization and nine for spin-pair correlations. The position of spin pairs in the correlations functions denoted by Ck ðk ¼ 1; 2; y; 9; denotes the number of correlation zone) are presented in Fig. 2b. In this approximation, the correlation functions of two spins laying farther than the radius of the outer (ninth) zone are decoupled and presented as the products of corresponding magnetization ms : The proposed approximation is based on the fact that the correlation between spins laying far from each other are weak with respect to the ones between spins separated by only a few atomic distances. It is worth to notice that the range of correlations taken into account in our calculations is equivalent to the third Matsudaira approximation [34]. The solution of this set of equations is obtained by iteration, where the equations for the correlation functions are treated as successive corrections taken into account in the basic equations for the surface and the bulk magnetization. The parameter l ¼ Jb =Js appearing in the coefficients bl and ak (see Appendix A) is connected with the distinction between the surface exchange integral Js for nn spins (both in the surface layer) and the bulk exchange integral Jb for all the other spin pairs. We assume that the value of this parameter can vary from 1 to 0. With respect to this parameter let us consider some particular cases:

35

0.3 λ = 0.4 0.2 0.1 0.0 0.0

(b)

0.5

1.0 kT Js

Fig. 3. The bulk (bold line) and the surface (thin line) magnetization versus temperature kT=Js for parameter l ¼ 1 (a) and l ¼ 0:4 and 0.5 (b).

bulk magnetization in the large interval of temperature. 2. 0olo1 ðJb oJs Þ means that the magnetic interactions between atoms laying in the surface layer is bigger than those between the bulk atoms. Fig. 3b shows the temperature dependence of the magnetization for the surface layer and for the bulk calculated for two different values of parameter l (0.4 and 0.5). Comparing the curves from Fig. 3b with these from Fig. 3a we observe a general enhancement of surface layer magnetization and, for instance, in the case l ¼ 0:4 this magnetization is remarkably bigger than the bulk one over wide temperature

I. Zasada / Physica B 334 (2003) 21–43

region. Respectively, for lower interactions the Curie temperature point is shifted toward the lower values. 3. l ¼ 0 ðJb ¼ 0Þ means that there are no magnetic interactions in the bulk. In this case, the obtained equations are reduced to the pure 2D problem and they reproduce the equations of Matsudaira [34] for z ¼ 4: This case is considered only for cheking the numerical procedure for the surface equations and it will not be considered further for the DSPLEED calculations.

0.2

λ = 0.4

Ck

36

0.1 k = 1 ... 9

0.0 0.0

7. Surface magnetism of Ni(1 0 0) by diffuse SPLEED Let us consider the magnetic properties of the topmost layer of the Nið1 0 0Þ surface of a single crystal. The intensity, spin–orbit and exchange asymmetry profiles have been calculated using the relativistic dynamical diffuse SPLEED formalism described in detail in Sections 2–5. The effective scattering potential is assumed to have the usual muffin-tin form. In constructing the

1.0 kT Js

1.5

0.05 0.04

In Fig. 4a the temperature dependence of nine correlation functions Ck ðk ¼ 1; 2; y; 9Þ has been presented for the case of l ¼ 0:4: As expected, the first correlation function (nn) is much bigger than the rest and plays the main role in the considered system. In Fig. 4b the distance dependence of spin–spin correlations at T ¼ Tc is shown. It can be seen that for very small inter-atomic distances the correlations are relatively strong and are vanishing quickly to the zero value for further spins. The behaviour of the correlation functions for other values of parameter l are very similar although their numerical values for a given temperature are different for each case. We now dispose of the mathematical tool, which allows us to determine the spin-pair correlation functions Gðp; p0 Þ ¼ /S0z Siz S  m2s necessary for the Fourier transform equation (31) calculations as well as we are able, at the same time, to have the bulk magnetization useful in the classical SPLEED calculations of Mc mk mk (see Eq. (24)).

0.5

(a)

0.03 0.02 0.01 0.00 a

(b)

2a

3a

4a

|i - j|[a]

Fig. 4. The temperature dependence of nine correlation functions Ck k for parameter l ¼ 0:4 (a) and the distance dependence of spin–spin correlation at T ¼ Tc ðl ¼ 0:4Þ (b).

scattering potential for a magnetic atom, an exchange potential with the average magnetic field representing the exchange field has been used. It means that the exchange potential is constant within the muffin-tin sphere and we have two scattering potentials (one for each of two arbitrarily chosen directions of the magnetic field). For each scattering potential the radial Dirac equation was solved numerically (Barbieri, Van Hove phase shift package [35]) to obtain the magnetic phase shifts with the spin–orbit effects included. The correlation functions determined in Section 6 will be now used to calculate the spin-dependent

I. Zasada / Physica B 334 (2003) 21–43

*

*

the first one determines the elements of the transition matrix in the basis of free spherical waves for the magnetic layer (Eq. (21)); the second one provides the spin-dependent scattering amplitude of Nið1 0 0Þ single-crystal surface (standard SPLEED technique); the renormalized spin-dependent transition matrix of the layer with magnetic disorder in the presence of the bulk is calculated in the third one (Eq. (32)) in which the input data are the output data of two previous programs; the last program allows us to calculate the spindependent diffuse LEED (DSPLEED) intensities of the magnetic system (Eq. (30)) and the incoherent scattering spin–orbit and exchange asymmetries (Eq. (6)).

This set of programs has to be used twice for two directions of the considered system magnetization (parallel and antiparallel relative to the surface normal). To exemplify the present theory we restrict the calculations to one fractional order position of a back-scattered beam i.e., ð0; 1=2Þ: All calculations are performed for normal incidence at T ¼ Tc i.e., in the phase transition point where the magnetic properties of the sample are entirely described by the correlation effects. At the Curie temperature the magnetization of the topmost Ni atomic layers decreases to zero and the exchange asymmetry connected with the coherent scattering processes disappears. The only scattering intensities which can be observed during the phase transition are connected with the critical phenomena occurring at the surface. Fig. 5 displays the results of the DSPLEED intensity calculations for the Nið1 0 0Þ surface. The top graph shows the spin-averaged diffuse intensity (i.e. ðImm þ Imk þ Ikm þ Ikk Þ=2) while the middle and bottom graphs present the resulting incoher-

diffuse intensity [arb. units] Aso [arb. units]

*

Aex [arb. units]

diffuse LEED intensities. For all calculations the geometrical parameters commonly found in the literature [9] are used. Calculation of the spindependent diffuse LEED intensities for the magnetic system requires the use of the following computer programs:

37

40

80

120

160

E [eV] Fig. 5. DI–V (top graph), Aso –V (middle graph) and Aex –V (bottom graph) profiles for (0, 1/2) beam at normal incidence from the Nið1 0 0Þ magnetic surface characterized by l ¼ 0:4:

ent spin–orbit asymmetry and the incoherent exchange asymmetry parameters, respectively. The physical information included in the intensity and spin–orbit curves is mainly connected with the geometrical parameters and these quantities are, similarly to the DSPLEED for non-magnetic materials [13,36], very sensitive, for example, to the interlayer spacing. As for the exchange asymmetry parameter, the observed effects are quite sizeable taking into account that it is coming only from the critical magnetic scattering at the surface. This parameter is expected to be sensitive to the changes in the correlation effects responsible for the surface magnetic properties in the critical temperature region. Fig. 6 shows the theoretical predictions for both the incoherent exchange asymmetry and the DSPLEED intensity for assumed changes in the correlation effects caused by taking different

I. Zasada / Physica B 334 (2003) 21–43

diffuse intensity [arb. units]

38

Aex [arb. units]

8. Final remarks

40

80

120

160

E [eV] Fig. 6. Aex –V (top graph) and DI–V (bottom graph) profiles for (0, 1) beam at normal incidence from the Nið1 0 0Þ magnetic surface characterized by l ¼ 0:4 (bold line) and l ¼ 0:5 (thin line).

values of parameter l describing the magnetic properties at the surface. The top graph presents the incoherent exchange asymmetry calculated for l ¼ 0:4 (bold line) and l ¼ 0:5 (thin line). The bottom graph provides the DSPLEED intensity for these two values of l parameter. As predicted, the magnetic surface information is exclusively coded in the scattering asymmetry. The spin average diffuse intensity spectra for both magnetic parameters l are indistinguishable. One sees clearly that the changes in the incoherent exchange asymmetry curve are not very big but they are evident and can be used for the magnetic structure analysis. The examples show that DSPLEED can be used to study the magnetic critical behaviour of the surfaces being sensitive to their magnetic characteristics at T ¼ Tc : Of course, similar calculations can be made also at the temperatures below Tc : In this case magnetization is not equal zero any more and the correlation effects are stronger and they lead to the well-ordered magnetic structure at the surface. In this way, the presented method is complementary to SPLEED and it can be very helpful for the magnetic surface properties analysis.

The critical behaviour of semi-infinite systems has been and still is the subject of great current theoretical interest. The presented considerations seem to be an interesting effort towards a systematic investigation of critical behaviour of magnetic surfaces. It is shown that by diffuse scattering of spin-polarized low-energy electrons, the information on the surface critical behaviour of ferromagnets can be also available in an original way. Up to the mid-1980s, LEED was considered as a surface structural tool applied only to surfaces with long-range order. It was then pointed out, first by theoretical work [37], that long-range order is not really essential. Diffuse LEED intensities, particularly those appearing in disordered adsorption on a crystalline substrate, carry information about the local adsorption structure. Different approaches have been proposed and put into practice to calculate the diffuse intensity distribution; for example, a cluster-like calculation [38], a theory of incoherent scattering with short-range order included [14], or the recent approach proposing the use of spin-polarized electrons [13]. The experimental techniques to measure and process the diffuse data were developed on the basis of the conventional LEED optics [39–41]. A number of disordered structures has been solved and reviewed in Ref. [42]. The interpretation of diffuse intensities brought also a new understanding of conventional LEED. Both DLEED and LEED intensities are created by the local scattering object and it is only the interference between the ordered and the disordered adsorption clusters that forms sharp spots or diffuse intensity distribution [43]. This local picture of electron diffraction led as well to a holographic interpretation of diffuse intensity distribution [44] allowing a reconstruction of real space atomic images (see Ref. [42] and the references therein). In this paper, we have reported a theory describing the low-energy back-scattering of spinpolarized electrons at a magnetically disordered surface of a semi-infinite monocrystal. The theoretical treatment presented here uses both theory and practice of the general DLEED theory as well

I. Zasada / Physica B 334 (2003) 21–43

as of the classical SPLEED theory keeping at the same time the originality within the context of the usual DLEED and SPLEED formalisms. The evaluation of the incoherent part of the DSPLEED intensity coming from the elastic diffraction on the magnetic surface is a new result which extends the use of the SPLEED theory and suggests the possibility to analyse the critical phenomena occurring at the surface during the phase transition. The model of a magnetic order at the surface allows us to determine the magnetic properties of the considered system. It is worth while to notice that the surface and bulk magnetizations determined in Section 6 can be also used in the classical SPLEED data analysis where the top-layer magnetization is estimated rather than calculated. The magnetic DSPLEED opens the possibility to get information about the layer, temperature- and field-dependent magnetization of the considered surface, especially in the region of phase transitions. From the theoretical point of view, the present theory is a relativistic extension of the DLEED. In the case of the magnetic order and the observations of its inhomogeneities the electron scattering should be sensitive to detect the magnetic crosssection distribution in analogy to the neutron scattering in the case of its diffuse nature [45]. It is worth while to notice that the neutron diffraction collects the data from the volume of the sample, while the electron diffraction allows us to analyse only the surface effects. Simultaneously, this study confirms the fact that the appearance of magnetic fluctuations in the vicinity of the Curie temperature can be analysed not only by the neutron investigations [46] but also by means of the diffuse electron scattering when the polarization of the electron is taken into account. Moreover, this confirmation concerns the surface magnetization distribution in its different sources of appearance, e.g., magnetic fluctuations, or ripple texture of domain structures. From the practical point of view, the present considerations give a tool for the analysis of relations between the electron scattering and the pair correlation function for magnetic moments in the surface region which is equivalent to the result for neutron scattering in the case of bulk systems.

39

The magnetic scattering is very weak and the experimental studies of the new structure have to be a challenging task. There are many aspects which have to be taken into account while performing the magnetic DSPLEED experiment. First of all, an experimental set-up allowing simultaneous probing of exchange and spin–orbit scattering is necessary [47,48]. The spin–orbit asymmetry, together with scattering intensity, reflects non-magnetic properties (like surface geometry and lattice vibration), while the layer and temperature-dependent magnetization is almost exclusively coded in exchange asymmetry if one employs the special diffraction geometry [49] with a sample magnetic field and incident beam polarization normal to the scattering plane. Of course, the magnetic structure has to be studied from exchange asymmetries, once the geometry was determined. The second very serious problem is connected with the distinction between the DSPLEED signal and the background signal. This takes place in particular because the inelastic or quasi-elastic scattered electron intensities are more or less homogeneously distributed over reciprocal space and they are often more intensive than the diffuse elastic signal [50]. A careful suppression of inelastic intensities is necessary, e.g., by means of an energy selective device, and a detector of high sensitivity is needed because of the fact that a very low signal has to be detected (about 1% of the specular beam intensity). However, based on the considerations in Ref. [51] one can conclude that such a signal is within reach of contemporary experimental techniques since half-order beams in this intensity range were indeed detected. Thus, the study of the diffuse background in a spin- and energy-resolved mode [52] should allow detection of the features explored theoretically in this paper. It is our hope that the present theory will stimulate new experimental studies of magnetic surfaces. Appendix A. The list of coefficients from Eqs. (49) and (50) b1 ¼

1 ½F6 þ 10F5 þ 44F4 þ 110F3 211 þ 165F2 þ 132F1 ;

I. Zasada / Physica B 334 (2003) 21–43

40

1 ½F6 þ 6F5 þ 12F4 þ 2F3  27F2  36F1 ; 29 1 ¼ 7 ½F6 þ 2F5  4F4  10F3  5F2 þ 20F1 ; 2 1 ¼ 5 ½F6  2F5  4F4 þ 10F3 þ 5F2  20F1 ; 2 1 ¼ 3 ½F6  6F5 þ 12F4  2F3  27F2 þ 36F1 ; 2 1 ¼ ½F6  10F5 þ 44F4 2  110F3 þ 165F2  132F1 ; ðA:1Þ

b2 ¼ b3 b4 b5 b6

where we have denoted n  Fn ¼ tanh b0 l 2

ðA:2Þ

and b0 ¼ Js =kT; l ¼ Jb =Js :    1 1 0 0 a1 ¼ 7 6 tanhðb lÞ þ 12 tanh b l 2 2 þ tanhðb0 ðl þ 1ÞÞ

 1 0 þ tanhðb ðl  1ÞÞ þ 2 tanh b ðl þ 2Þ 2   1 þ 2 tanh b0 ðl  2Þ 2   1 0 þ 4 tanh b ð2l þ 1Þ 2   1 0 þ 4 tanh b ð2l  1Þ 2   1 0 þ 8 tanh b ðl þ 1Þ 2   1 þ8 tanh b0 ðl  1Þ : ðA:3Þ 2    1 1 a2 ¼ 7 6 tanhðb0 Þ þ 12 tanh b0 2 2 0

þ tanhðb0 ðl þ 1ÞÞ



  1  tanhðb0 ðl  1ÞÞ þ 4 tanh b0 ðl þ 2Þ 2   1 0  4 tanh b ðl  2Þ 2   1 0 þ 2 tanh b ð2l þ 1Þ 2   1 0  2 tanh b ð2l  1Þ 2



 1 0 b ðl þ 1Þ 2   1 0 8 tanh b ðl  1Þ : 2    1 1 0 a3 ¼ 5 6 tanhðb Þ  12 tanh b0 2 2 þ 8 tanh

ðA:4Þ

þ tanhðb0 ðl þ 1ÞÞ   1 0  tanhðb ðl  1ÞÞ  4 tanh b ðl þ 2Þ 2   1 þ 4 tanh b0 ðl  2Þ 2   1 0  2 tanh b ð2l þ 1Þ 2   1 þ 2 tanh b0 ð2l  1Þ 2   1 0  8 tanh b ðl þ 1Þ 2   1 0 þ8 tanh b ðl  1Þ ; ðA:5Þ 2    1 1 0 0 a4 ¼ 5 2 tanhðb lÞ  4 tanh b l 2 2 0

þ tanhðb0 ðl þ 1ÞÞ

  1 0 þ tanhðb ðl  1ÞÞ þ 2 tanh b ðl þ 2Þ 2   1 þ2 tanh b0 ðl  2Þ ; ðA:6Þ 2    1 1 0 a5 ¼ 5 2 tanhðb Þ  4 tanh b0 2 2 0

þ tanhðb0 ðl þ 1ÞÞ  tanhðb0 ðl  1ÞÞ þ 2 tanh 



 1 0 b ð2l þ 1Þ 2

 1 0 b ð2l  1Þ ; 2    1 1 a6 ¼ 5 6 tanhðb0 lÞ  12 tanh b0 l 2 2 2 tanh

þ tanhðb0 ðl þ 1ÞÞ

ðA:7Þ

I. Zasada / Physica B 334 (2003) 21–43

  1 þ tanhðb0 ðl  1ÞÞ  2 tanh b0 ðl þ 2Þ 2   1 0  2 tanh b ðl  2Þ 2   1 0 þ 4 tanh b ð2l þ 1Þ 2   1 0 þ 4 tanh b ð2l  1Þ 2   1 0  8 tanh b ðl þ 1Þ 2   1 0  8 tanh b ðl  1Þ ; ðA:8Þ 2

a7 ¼

  1 þ tanhðb0 ðl  1ÞÞ  2 tanh b0 ðl þ 2Þ 2   1 0 2 tanh b ðl  2Þ ; ðA:11Þ 2 a10 ¼

 1 0 þ tanhðb ðl  1ÞÞ þ 2 tanh b ðl þ 2Þ 2   1 0 þ 2 tanh b ðl  2Þ 2   1 0  4 tanh b ð2l þ 1Þ 2   1 0  4 tanh b ð2l  1Þ 2   1 0  8 tanh b ðl þ 1Þ 2   1 0  8 tanh b ðl  1Þ ; ðA:9Þ 2 0

  1  tanhðb0 ðl  1ÞÞ  4 tanh b0 ðl þ 2Þ 2   1 0 þ 4 tanh b ðl  2Þ 2   1 þ 2 tanh b0 ð2l þ 1Þ 2   1  2 tanh b0 ð2l  1Þ 2   1 0  8 tanh b ðl þ 1Þ 2   1 0 þ8 tanh b ðl  1Þ ; ðA:12Þ 2



a11 ¼

  1 þ tanhðb0 ðl  1ÞÞ  2 tanh b0 ðl þ 2Þ 2   1 0  2 tanh b ðl  2Þ 2   1 0  4 tanh b ð2l þ 1Þ 2   1 0  4 tanh b ð2l  1Þ 2   1 0 þ 8 tanh b ðl þ 1Þ 2   1 0 þ8 tanh b ðl  1Þ ; ðA:13Þ 2

 1 0  tanhðb ðl  1ÞÞ  2 tanh b ð2l þ 1Þ 2   1 þ2 tanh b0 ð2l  1Þ ; ðA:10Þ 2 0



   1 1 0 0 a9 ¼ 3 2 tanhðb lÞ þ 4 tanh b l 2 2 þ tanhðb0 ðl þ 1ÞÞ

   1 1 6 tanhðb0 lÞ  12 tanh b0 l 2 2 þ tanhðb0 ðl þ 1ÞÞ

   1 1 0 a8 ¼ 3 2 tanhðb Þ þ 4 tanh b0 2 2 þ tanhðb0 ðl þ 1ÞÞ

   1 1 0 0 b 6 tanhðb Þ þ 12 tanh 23 2 þ tanhðb0 ðl þ 1ÞÞ

   1 1 0 0 b 6 tanhðb lÞ þ 12 tanh l 23 2 þ tanhðb0 ðl þ 1ÞÞ

41

a12

   1 1 0 ¼ 6 tanhðb Þ  12 tanh b0 2 2 þ tanhðb0 ðl þ 1ÞÞ

I. Zasada / Physica B 334 (2003) 21–43

42

 tanh ðb0 ðl  1ÞÞ  4 tanh   1 0 þ 4 tanh b ðl  2Þ 2   1 0  2 tanh b ð2l þ 1Þ 2   1 þ 2 tanh b0 ð2l  1Þ 2   1 0 þ 8 tanh b ðl þ 1Þ 2   1  8 tanh b0 ðl  1Þ : 2

  1 0 b ðl þ 2Þ 2

ðA:14Þ

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