Spin polarized low energy electron diffraction (SPLEED)

Physica 142B (1986) 332-347 North-Holland, Amsterdam

SPIN POLARIZED LOW ENERGY ELECTRON DIFFRACTION (SPLEED) K. STACHULEC Politechnika Swietokrzyska, Aleja lO00-Lecia P.P. 7, 25-314 Kielce, Poland Received 1 July 1985 Revised maquscript received 16 May 1986

A simple dynamical approach to the description of the Spin Polarized Low Energy Electron Diffraction (SPLEED) by a solid state sample with a free surface is given. The scattering amplitude for a given polarization of the incident electron beam is expressed in terms of the mean square displacements of atoms and the electron density distribution at a surface of the scattering thin film sample. The obtained formulae for the scattering potential and for the scattering amplitude in special cases lead to those obtained in the literature.

1. Introduction In early Low Energy Electron Diffraction (LEED) experiments the spin direction of the scattering electrons was seldom considered. The spins in electron beams that were produced by thermal emission had arbitrary directions. Whenever the spin direction played a role, one had to average over all spin orientations in order to describe the scattering properly. Only in recent years it has been possible to produce free electron beams in which the spins have a preferential orientation. They are called polarized electron beams and at present they are more frequently used in LEED experiments as well as in theoretical descriptions of these experiments. In the present paper we give a simple approach to the description of the Spin Polarized Low Energy Electron Diffraction (SPLEED) by a solid state sample with a free surface. The sample is treated as a thin film evaporated on a substrate. The sample is treated as a thin film evaporated on a substrate. We understand a thin film as a system of few monatomic layers parallel to the surface of the sample while the other part of the sample is treated as the substrate. For such samples we have calculated a scattering potential including the exchange of the incident electrons with the electrons of the sample. The obtained potential is a temperature and thickness dependent quantity. In the next step, starting from the Dirac equation we have calculated the scattering amplitude in the first Born approximation. The formulae obtained for the SPLEED potential and thus the SPLEED amplitudes are exact and they lead as special cases to those obtained for LEED earlier.

2. The scattering potential for spin polarized LEED We shall assume that a spin polarized low energy electron beam is incident on a perfectly clean, well-ordered surface of a sample. Our interest will focus on the elastically scattering electrons, because they produce almost all the structure in the diffraction patterns. To construct the scattering potential we divide the scattering sample into a thin film and a substrate. By the thin film we understand n monatomic layers parallel to the surface which are numbered by v, starting with v = 1 for the free surface of the filmand finishing with v = n for the atomic layer evaporated directly on the substrate. Any monatomic layer v is divided into two-dimensional elementary cells and the position of any cell inside the vth atomic layer related to the cell chosen as the origin of the coordinate system is given by a 0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

K. Stachulec / Spin polarized low energy electron diffraction •v

333 ,p

-v

two-dimensional vector j~ = aj~ + bjy, where a and b are lattice vectors and I~, Ir denote integer numbers. Let the z-axis be perpendicular to the surface and directed into the inside of the thin film, a position of any atom in the film can be described by the vector

R~j~ =j~ + d~c~v + p~----R~j~ + p~ ,

(2.1)

where p~ describes the position of an sth atom belonging to the (vj,)th cell related to the origin of the local coordinate system bounded with the (vj,)th cell inside the thin film. From the diffraction problem point of view the thin film is regarded as a system of n • N 2 bounded atoms, where N z denotes the number of atoms in any vth layer parallel to the surface. Such a system produces a suitable effective scattering potential localized round about each lattice node of the film. In the general case the scattering potential of the (vj,s)th atom of the thin film depends on the position ( r - R,j.,) and spin coordinates s o of the diffracted electrons. Denoting this potential by V(r - R~j,,, So) , a total scattering potential on the position r inside the film V(r, So) can be treated in the first approximation as the superposition of the effective potentials of the individual atoms from which the film is built:

V(r, So) = ~ V(r - R~j~s , So).

(2.2)

vjvs

By the effective potential of an individual atom we understand the potential produced by a given atom in the presence of other atoms of the thin film and the substrate on which the film is evaporated. This potential can be constructed by some modification of the scattering potential of a free atom. Now, let us consider the scattering of an electron by a free atom. Denoting by r 0 and s o the position and spin coordinates of the scattered electron, and by rls,, r2s 2 . . . . , rNS N the position and spin coordinates of the atomic electrons, where N denotes the total number of electrons in the atom, we should describe the scattering problem by means of a wave function ~ = q)(roSo, rls z . . . . , rNSN) which depends on all coordinates of the (N + 1) electrons. The many-body function ~ has to satisfy an equation which we write in the form {j_~o ( --

h2

--~m V2-

Ze2) N rj / + Z j = O E i=j+l

e2

}

iriTrj I ~=Et(~ ,

(2.3)

where -(h2/2m)V 2 is the operator of the kinetic energy of the jth electron, so that E~=0 [-(h2/2m)V~] describes the kinetic energy of the system of N atomic electrons plus an incident electron. The term E~=0 (-Ze2/rj) is the potential energy of all electrons in the field of nucleus, the third term in eq. (1.3) stands for the Coulomb interaction between electrons. E, denotes the total energy of the system and Ze the nuclear charge, while j is the number of electrons of the system. The difficulty in eq. (2.3) is that electrons influence one-another via the Coulombic repulsion and the incident electron may distort wave functions of the atomic electrons by its own electrostatic field, correlating their motion with its own and changing the effective potential seen by the incident electron itself. If the free atom wave functions of the electrons 01(rlsl)~2(r2s2)... qju(rNSu) are known, we can in some approximation express the many-body function ~ by means of the free atom functions as follows I2]: = K

*N(rN

N),

where $ ( r j o ) is the incident electron function.

(2.4)

K. Stachulec / Spinpolarizedlow energyelectrondiffraction

334

The sum is over permutations of the particle coordinates and ep takes the value of + 1 if permutation p can be achieved by exchanging an even number of particle coordinates, and - 1 for an odd number. There are (N + 1)! possible permutations. Since the atom state part of • is known, we want to eliminate this part from our equations and concentrate on (a(roSo)being the incident electron wave function. Multiplying eq. (2.3) by X *=

@l(rlsl)@2(r2s2)... @N(rNSN),

(2.5)

integrating over space, and summing over spin coordinates of X we obtain the equation for qb(r0s0):

h2

Ze 2

ro

+

e2~ f ~ Ilfij(rjsj)12d3rj]qb(roSo) Vo:

-e2 ~ [f ~ qJT(rjsj)q~(rjsj)-~o~-~i I d3rj] ~bj(roSo)= Eoqb(roSo),

(2.6)

where E 0 denotes the energy of the incident electron. The last equation is a one-electron equation for the wave function of the incident electron and it defines the effective scattering potential of the free atom Vo(roSo):

Vo(roSo)= -ro- - e E j where by

Vexc(roSo)we

j(rjsj)[

0-V,I d3r' + V°x°(r°' s°) '

(2.7)

denote the so-called exchange potential, which is defined by the equation

Vexc(roSo)C~(roSo)= . •,,

f d3rj7"(rjsj)6(rJsJ) ~o--~-~j] ~bj(roSo).

(2.8)

The two first terms on the right-hand side of eq. (2.6) have a simple interpretation as the electrostatic potentials produced by the nucleus and the electron density distribution of the atom, respectively. The exchange of potential defined by (2.8) is non-local potential and it arises out of considerations of antisymmetry under exchange of particles; because no two electrons can be in the same place at the same time hence each electron is surrounded by a region depleted of other electrons and hence lower potential. Writing the one-electron wave functions d/j(rjsj) as tpi(rjsj)= ~bj(rj)xj(sj), where Xj(si) is the spin coordinate dependent part of the wave function of jth atomic electron while Oj(rj) is only space coordinate dependent, we can express the electrostatic potential of atomic electrons by means of the electron density distribution of the atom:

e ~• Z', f Illlj(rjSj)12 ~o_--r-~fl d3r/= -e j~ f d3r ]rDJ(r) °- rl.

(2.9)

Analogically, writing ¢k(roSo)= ~b(ro)~(So) where ~(So) is the spin wave function of an incident electron, the exchange potential can be written as

K. Stachulec / Spin polarized low energy electron diffraction

V (roSo)

J

In( 0)l

335

Ij(ro)

=~]6sosi~(ro),

(2.10)

J

where

Ij(ro) =

~bj(ro)~b*(ro)

14,(ro)l 2

f d3rj ~/'7(ry)~b(ry) [r° _ rYI

(2.11)

and the sum is over the atomic electrons which have the same spin as the incident electron. In this way a full scattering potential of a free atom takes the form

Vat(roSo ) = --r Ze

- e ~•

f

- -pj(r) oJ r- r[ d3r + ~]j 3~o~jlj(ro).

(2.12)

The above formula will be a basic to construct the scattering potential for the polarized electrons by thin films by means of the relation (2.2). It is known that crystal lattice atoms at any temperature take part in the temperature vibrations around their equilibrium position and the influence of the temperature on the scattering potential must be taken into consideration defining a dynamic scattering potential VT(r, So) for an atom as [9]

VT(r , So) = f

Vat(r;

s~)T(r - r')

d3r ' ,

(2.13)

where T(r) describes a distribution of the mass center of an atom in its vibration around the equilibrium position. For the atoms bounded in a crystal T(r) is determined by the lattice dynamics. A n o t h e r point we would like to stress is that placing an atom in a lattice site of an infinite sized sample does not change the mean number of electrons per atom. If the sample is infinite, the mean square displacement which characterizes the dynamics of the atoms in a sample is the same for every atom. However, in the thin crystal case the surface atoms of any solid body are in a situation which is different from that of atoms situated in the inside of the film. The surface atoms feel the changes in the geometry of the neighbours surrounding them, caused by the missing neighbours, by the spontaneous deformation of the lattice near the surface. As a result, the scattering potential near the surface must be different from that inside of a bulk material. It is to be expected that the changes of the electronic structure near the surface must cause some changes of the physical properties related to the surface. To take the above into consideration we introduce the effective numbers of electrons per jth orbital of the J (ujvs) atom in the film (nvjvs), which are the same for the atoms in the vth monoatomic layer but they create a distribution in the direction perpendicular to the surface. The redistribution of the electronic charge in the thin film creates some new boundary conditions for lattice vibrations which must influence the temperature dependence of the mean square displacements of the atom in the other atomic layers parallel to the surface. Denoting the mean square displacement of the (uj~s)th atom in the thin film by Bvj~, we can write (2.14) as well as for the electron density distribution: J (nvj~s) = (n~s) ,

(2.15)

336

K. Stachulec / Spin polarized low energy electron diffraction

where the symbol ( - . . ) stands for the thermodynamical average, and the above relations are the consequence .of the translation symmetry of the film. By means of the above dynamical parameters Bvs and (nt~) we propose the following modification of the scattering potential of the atom in the (vj~s)th site of the film: VT(r o -- R,j,s, So) =

dr'(,'at(r' - R,j.s, So) exp

2B~

'

(2.16)

where

2B~s describes the distribution of the mass center of (vj~s) atom in its temperature vibrations [9] and in place of the free atom potential Vat given by (2.12) we take l?at which we define for (vj~s)th atom as

V~t(ro-R~vs, So)=

Ze Iro -

R~j~sl

e ~t (n~s) f pt(rj -Rvj~s) d 3 r t + e ~ , (nt~)Ij(r°-R~j~s)~'~°" ~o---~]yl (2.17)

To obtain an analytic formula for the scattering potential of the spin polarized electrons scattered by thin films, using the formulae (2.2), (2.16) and (2.17), we have to know the atomic orbitals of a free atom by means of which the quantities Pt and It appearing in formula (2.17) are expressed. For that we restrict our next considerations to the case when the free atom orbitals are spherically symmetric so they can be taken in the Slater form [3, 10]

~bt(r) = Air "j e -xjr ,

(2.18)

where /zt, At are the numerical parameters given for all free atoms in ref. 10 and Aj is a normalization constant. If we now write the incident electron wave function in the form of the plane w ave: th(r) = eik" ,

(2.19)

where k is the wave vector of the incident electron, we can insert expressions (2.18) and (2.19) into eq. (2.17) to calculate the static scattering potential, ~'at(r0 - Rvjvs , s0), of (vjvs)th atom of the film. We have

Ze - 4zre ~ Aj(n~s 2 j ) l)'~t(r° - R"i"s' So) = ]ro - R~j~] i [ro-R ~i~l ×

o

Iro -

- R Jro-R ~j~l

(Irj

~,1) 2'~+2

1)2~,+l exp(-2A]lrj - evj~sl) dr]]

K. Stachulec / Spin polarized low energy electron diffraction

+ 4ere E A~(nJ~)lro - R~j~,I "j exp{-(h0 + ik')lr0 J

337

R~j=s[}

[ro-Rvy, sl

1

[Iro-R~v~l + f Iro - r

f

([rj - R~j~[) "j+2 e x p { - ( h j - ik')lrj - R~j~,I} drj

o

([r-R.~l)~'+lexp(-(A,-ik')lrj-R...l}drj]

•

(2.20)

~j~s[

Integrating over rj by means of the standard formulae [11] u

f x v-1 e -~x dx = I*-"Y(v, lxu) , o oo

f x ~-1 e -~'x dx = Iz -~F(p, ~ u ) , u

where y(v,/~u) and F(v, tzu) are the so-called incomplete gamma functions, we obtain

Ze

¢'at = Iro_R~j.s[

47re E A~. j

(nJ~')

(2Ai)2------~, +2

x [ y((txj + 3). It0 - R~j.,I2Aj) Ir0 - R~i.~I2Aj + F(/zi + 2, [to - R.i~I2Aj) ] + 4¢re ~. Aj3~,~o(n~ 2 j } exp(-(Ai + ik')lr o - R,,j~I) 1

×

Ir0 R"J.'I~'J[[ y(t~j + 3, (Aj - i k ' ) l r o - R#~I) l

Iro - R~j~, [('~j - ik')

+ F(/xj + 2, (Aj - ik')lr 0 - R~i.s[) ] (Aj - ik') -°'j+2) .

(2.21)

Next, the calculation for the dynamical scattering potential by means of the formula (2.16) and explicit form of ¢'at given by (2.21) leads to the expression ir VT(r-R~J~'s°)=exp

where

g

2

~

2k

----.j~[ V T (B,,s, So) , 2B,,~ ] k~.~ =° (1/B,,~) ( 2 k + 1 ) ! ( [ r - R~,~[) 2k-(k)

(2.22)

K. Stachulec / Spin polarized low energy electron diffraction

338

1 ~3/2;47rr,2k+2exp( v(k)( B~s' s°) = (\ 2---~/

r ' 2 -)Vat(r ,' so) d3r' 2Bv~

0

= Ze(Fo,_a)k - E 4~reA2(n~) (2/zj + 1)! j

(2hi) 2~'j+2 2p,j + 2

x { 2+~2i2Aj _

(F°'-X)k -- (2/zi + 2) m=o ~ (2hi)m-' m! (F2xj'm-1)k

2~j+, (2Aj)m } 2 j (2/zj + 1)! + m=o ~ m! (F2xpm)k + ~j 4~reAj3sj~o(n~) fl~j+2 /zj+2

(/z~+2) flj ( t3,,~,-,)k

-

~

m =0

m-1

P7 + 1

m

m! (Ft3'"m+~'-l)k + ~'~ ~fli. v

(F~,,"+-)

k

'

m=0

(2.23) where Fxn =-r" e -xr

and

(Fx,n) k stands for the following integral for the (vj~s)-site of the lattice

(Fx,n)k=(2~vs)3/2 f 4~r2k+2exp(-- r2 ) Fx,(r ) dr ' 1 )3/2 =4 (2--~ (B us)(2k+,+3)/2( 2 k + n + 2 ) ! e ~s.~,2~ " x/B-~__". I~_(2k+~+a)~XVn~D

(2.24)

In the last expression D,(x) denotes the standard function of the parabolic cylinder [11], while the quantity flj which appears in (2.23) is defined as

[3j=_2hj _ ik(l _ cos O) = 2( Aj - ik sin 20)

(2.25)

and 0 is the scattering angle of the electron. The final result for the total scattering potential produced by thin films for the spin polarized electrons takes the form

VT(r, s0) = E exp

([r-RvJ~s[2) ~

vj r s

k~=0(1/B~s)2g (2k + 1)!

~/r(k)(l~ SO) --T \~vs'

(2.26)

=

and ("(Tk)(B~s,SO) is given by expressions (2.23), (2.24) and (2.25). This form of the scattering potential has interesting properties. At first this potential is finite at every lattice point R ~j~ contrary to the effective potential of the free atom which is infinite at the middle of the atom. Another thing we point out is that one can obtain from it the forms used up to now in the literature [7, 8]. Namely, if we ignore the exchange part of this potential and restrict the consideration to the high temperature limit, which means the high values of B~, so that the formula (2.26) can be approximated for simple cubic lattice by VT(r) ~ ~~j exp

( [r-Rvjl2] V(TO)(Bv) 2B~

/

(2.27)

339

K. Stachulec / Spin polarized low energy electron diffraction

where V~°)(B~) is given by

V~O)(B~) = Z e ( F o _ ~ ) o _ ~ 4zteAj 2(n~s) j (2/x~+ 1)! j (2A j) 2~j+2 2t~]+2 2/xj+1 2A---~ ( F ° ' - l ) ° - (2/.~j + 2 ) ~'. (2Aj)"-' (2Aj)" m=0 m! (F2~j 'm-l}° + m=0Zm----~ (F2~j 'm}°

(2p~j + 1~! =4,n'e ~ A~ (niv) (-~i~-f~,7~ 2.u,j+ 2 (2A j) m-1

× (2/.tj+2) ~, m=0

m!

2,o7+1 (2Aj)m (F2xj'm-1)°-]-

~

m=0

m----~. (F2'~j'm)°

}

(2.28)

and

(F2%,,) o = 47r ( 2 - -1~ )

3/z (Bv)(n+s)/2(n+2)' e ~ ' ~ D _ (,,+3)(2AjV'--B-:~).

(2.29)

The last two formulae in the hydrogen atom case, for which the wave function has the simplest form, lead to the result of Dworiankin [9]. Instead of the Slater-type function by means of which is calculated the electrostatic potential of free atoms, one can use the electrostatic potential of a free atom in the form [12] V(r, = - r

~--

aA e

r

,

where a~ and b~ are numerical parameters given in ref. 12. In this case the formulae (2.27)-(2.29) lead to the results we have obtained in the papers [4, 8].

3. Low energy electron diffraction amplitude including spin In analogy to the case of light beams, electron beams can be polarized by scattering, and the angular distribution of scattered electrons depends on the state of polarization of the incident beam. The effects can be treated by means of the Dirac equation, which is the basic equation for the electron including spin and its relativistic behavior. To calculate the scattering amplitude for the elastic scattering of an electron beam with arbitrary spin direction for LEED experiments we utilize the scalar surface scattering potential V.r(r, So) described in the previous section (see formulae (2.26)), which is a spin polarization dependent quantity. The problem to be solved is the Dirac equation including the potential V.r(r , So) , and we write it in the form e

( 4/V- ~/4K + k°)O(r) = hc y4Va'(r' s°)~b(r) '

(3.1)

where K = V ~ + k~, k 0 = mc/h, m stands for electron mass, E = chK for its energy, c for light velocity, and h for Planck constant, while by k we denote the value of the electron wave vector k. The matrices ~ = (Yl, 3'2, 3"3) are given by the Pauli matrices o-i [1]

340

K. Stachulec / Spin polarized low energy electron diffraction

"~.=-i/3a,,=

idr

0

,''

n=1,2,3

and

(,o) (1o) o

where I -

0

i'

1

is the unit matrix. The anticommutation relations for g/th matrices are

Tt, T,' + 4/,~,, = 2 8 ~ , , ,

/x, /,' = 1, 2, 3, 4.

(3.2)

Because ~, are 4 x 4 matrices, eq. (3.1) makes sense only when q,(r) has the form

ii/~'4/

33,

Then (3.1) represents a system of four simultaneous first order partial equations which must be solved with an asymptotic form e ikr

~b

> u e ikr + F(r~) - r-~

(3.4)

r

for the four components of the wave function ~b(r). To solve the system of eq. (3.1) with the boundary conditions (3.4) we use the Green function method. It allows one to write the system of the differential equation as the system of the integral equations Ill(r ) = U e ikr --

ehc f G(r, /")'~4VT(lr')l//(r

') d3r ' ,

(3.5)

where dp = u e ikr is the solution of the system of homogeneous differential equations: ((9 . V ) - g/4K + k0) 6 = 0

(3.6)

while the four-component spinor Green function G(r, r') satisfies a system of equations ( ~, . fT) - 94K + k o ) G ( r , r') = 8(r - r') .

(3.7)

Taking into account the relation ( . ~ . ~ - ~/4K -t- k 0 ) ( . ~ . V - .~4K- k0) --- AA- g 2 - k 2 ,

(3.8)

which can be easily obtained by multiplication and utilization of the anticommutation relations (3.2), one can look for the solution of (3.7) in the form G ( r - r') = ( ~, . V -

~4K - k o ) D ( r -

r') .

(3.9)

K. Stachulec / Spin polarized low energy electron diffraction

341

By applying (3.9) to eq. (3.7) one obtains an equation for D ( r - r'): (A + k 2 ) D ( r - r') = 8(r - r') ,

(3.10)

which can be solved as

D(r-

r') =

1

e ik[r-r'l

4Ir

Ir- r'l

hence the Green spinor G ( r -

(3.11)

' r') is given by

1

e iklr-r'l

G(r - r') = --~

( ~ . ~7- "y4K

1

-- ko) -]"r: ~ l e iklr-r'l

(3.12)

= 4---~('Y4K- (~ .~r) + k0 ) Ir _ r'[ "

The boundary condition (3.4) for the spinor ~(r) will be fulfilled if we take the asymptotic form of (3.12):

e ikr

1 G(r-

r')r~ ° ~

(Fa g - i k " ~ + k0) e -ik'r'

(3.13)

r

where k' is the wave vector of the electron at infinity. Now we can write eq. (3.5) utilizing (3.12) as ~b(r) = ~b(r) = qb(r)

e f e ikr 4zrhc dar ' (~4 K - i(k'. ¢/) + ko) e'k'r'~/4VT(r')~b(r) r'

e

41rhc ( Y 4 K - i k ' . ~ + ko)~4

f

e-ik'r'VT(r')C,(r ') d3r ' .

(3.14)

Solving this equation by iteration starting with ~b(r) = ~b(r), in the lth step of the iteration one obtains lth Born approximation of the solution: ~/(l)(r)

= t~(r)

--

e

47rh-----~(Y4K - i k ' . ~ + ko)~4

f

e-ik'r'VT(r')~l(l-1)(r ') d3r ' .

(3.15)

In a special case for l = 1 we have the first Born approximation of the spinor ~, which describes the scattering of the electron by the potential VT(r , So): ~b(1)(r) = u e i*"

e -- 4~rh----~ (Y4K - i k ' . ~ + ko)~4

f

e ikr d3/' r

e-i(k'-k)r'VT(r)u - -

(3.15a)

where we used the relation ~b = u e ik'. Now, the comparison of the asymptotic form of the spinor ~b (eq. (3.4)) with its in the the lth Born approximation computed form (eq. (3.15a)) leads to the lth Born approximation of the spinor amplitude of the outcoming spherical wave: F(O(rl ) = - 4~rh-----~ e (~4K - i(k'. ~) + ko)~4 f e-ik'r'VT(r')~b(l-1)(r' ) d3r '

(3.16)

342

K. Stachulec / Spin polarized low energy electron diffraction

and in the first approximation it takes the form

F(1)(r~)

_

e

4zthc (yaK - ik'. ~ + ko)4/4u

e-i(k'-~)r'VT(r' ) d3r.

(3.17)

The spinor u(tl) describing the incident electron, by which is expressed the spinor F(r~) in the two last equations, satisfies the following equation: (i;/- k - ~/4g -~- ko)u = O .

(3.18)

This allows one to conclude that

fi( ~4 K - i k . ~, + ko) = 2kofi ,

(3.19)

where fi= u+~, and plus denotes the Hermitian conjugate. Taking into consideration the above conclusion, the spinor F(°(r~), eq. (3.16), can be expanded over the polarization states of a free incident electron which are the solution of (3.18) and which we denote by ux(~), using the symbol A for two possible polarization states and h for the unit vector ( k ' / k ' ) = ~i. This can be written as F(°(r~) = ~'~ u,(rl)(fia(r~)F (t) (r~)) = ~ u,(ri )f(')(r~, A) A A

(3.20)

and one can call f(l)(Ft, A) the scattering amplitude for the given polarization of the incident electron beam. From (3.16) and due to (3.19) the scattering amplitude for a given polarization is given by

f(/)(rl, A ) =

me 2 Ifi~(tl)y4 f e-ikr'VT(r ', s ' ) ~ ( r ' ) d3r'l 2~'h

(3.21)

and in the first Born approximation, where ~0 = u(~0)e ikr, it takes the form f(r~, A) = fix(ft)y4ux(fto)f~o(k'-

k),

(3.22)

where we have omitted the symbol of the first Born approximation, because from this stage onwards we are going to discuss this case of approximation only. In the expression (3.22) we have introduced the quantity

f~o(k' - k) =

me f e_i(k,_k)r,VT(r,SO)d3r,

27rh 2

(3.23)

which corresponds to the nonrelativistic case of the LEED amplitude and describes the polarization effect of the scattering due to the exchange part of the scattering potential. If an incident beam is not polarized and atoms of the scattering sample are nonmagnetic, then due to the same values of the exchange potential for both directions of the incident electron spins, the amplitude (3.23) does not depend on the spin. In this case, for the magnetic sample, however, we have two different values of the exchange potential for different spin directions of the incident electron beam, and thus the two amplitudes are different. We can say the same in the case of the scattering of the polarized beam by the magnetic sample. We are now in a position to say that the relativistic scattering amplitude for a given spin polarization of the incident electron beams expresses itself by means of the nonrelativistic one, multiplied by the relativistic factor u~(r~)u~(ri0) created by the free electron spinor u for the incident

K. Stachulec / Spin polarized low energy electron diffraction

343

and scattering directions, respectively. To calculate this factor one has to solve eq. (3.18) and the values problem: ~b = s~b,

(3.24)

where s denotes the projection of the total angular m o m e n t u m J = r x p + ½hdr on the direction of the m o m e n t u m ri = t~/P and it is given by g = drp, because Jp = 0, and (3.24) can be written as

(dr. k)u = k s u ,

(3.25)

where we have used the plane wave form of ~b and have to write k / k instead of p/p. Eqs. (3.18) and (3.25) as a consequence of (3.3) represent a system of eight simultaneous algebraic equations which we write as

(sk - k~)u I = (k x - i k y ) u 2 ,

( e E - ko)u 1 = s k u 3 ,

(sk + kz)U 2 = (k x + i k y ) U l ,

( e E - ko)u 2 = s k u 4 ,

(sk - kz)U 3 = (k x - i k y ) u 4 ,

(eE+

(sk - k z ) u 4 = (k x + i k y ) U 3 ,

( e E + ko)u 4 = s k u 2 ,

ko)u 3 =

sku 1 , (3.26)

where we have put K = eE (E = V ~ + k~, k = (k x, ky, kz) ) and e is to be determined. All the equations (3.26) can be satisfied simultaneously if we put

(ul) zi1) u2

1 A 1 Bz

U----- U3 = ~ / A 2 nl u4 \ A 2 B2

.

In this way one can simply obtain the possible values of s and e. The calculations show that (3.26) have a solution only if both s and e take the value of --- 1. In the scattering p r o b l e m of electrons we are interested in the solutions with positive e ( = + 1 ) for which the spin s can take both the values, - 1. The two solutions in the spherical coordinate system after the normalization Esu~+ ( n~) u , ( n ) = 1 are (E = K)

1(

s~/(1 + ko/K)(1 + s cos 0 e -½i6

~/(1 + ko/K)(1 - s cos 0 e ½i*

us(ri) =

(3.27)

~/(1 - ko/K)(1 + s cos 0) e -½i~

s~/(1 - ko/K)(1 - s cos 0) e ½i6 where 0 and ~b denote the spherical angles of the unit vector ri, which describes the direction of the electron propagation. Denoting by rio the incident electron direction and letting it to be ri o = (1, 0 = 0, ~b = 0) we can describe the incident electrons by means of the spinor

C1 u ( rio) = - , ~

C 1 X,/1.__okO/ K j + --V~

X/1

7 -

~/1

°/ K -

ko/K /

,

(3.28)

344

K. Stachulec / Spin polarized low energy electron diffraction

where C 1 and C_~ are the normalization constants and now 0, tk denote the scattering angles, and we can calculate the scattering amplitude for a given spin polarization. We have (A = s) f(a, s) = ,,:

_

- !,)

C,

½i~"

[s(l+-~)+(l--~)]~/l+scosOf,(k'-k)e

+ ~ 2 [ ( l + ~-~)-s(l- ~)]~/l-scosO f _l(k'-k)e -½i* rC 1 cos ~0 e½i*fl(k ' -

ko

0

k) + C 1 -~ sin ~ e-½i*f_l(k' - k ) ,

Ko 0 0 - C 1 - ~ sin ~ e½i4'fl(k - k') + C_ 1 cos ~

(3.29)

e-~i°f_l(k' - k),

where the last two lines represent the scattering amplitudes for the s = 1 and s = - 1 polarizations of the scattered beam, and f+_(k'-k) is written to point out the fact that these nonrelativistic quantities can differ between each other in the cases under consideration. Eqs. (3.29) show that for every polarization of the diffracted beam we have two components of the scattering amplitude: one component describes the scattering process without changing the spin direction during the scattering and the second one corresponds to the scattering with a change of the spin direction. Now eqs. (3.29) allow us to compute the full scattering cross section for both polarized and non-polarized incident electron beams. The result is (f_+~ - f+_~(k' - k)) dcr = E d12

If(r~

.s)l

([Clfll 2 + 1 C _ l f _ l l

cOs2 ~ + k / k / ( [ C - l f - l l 2 --]Clfl[2) sin2 g

=

iCl[2+ iC_xl2

'

where the condition IC112 + I f _112 = 1 characterizes the polarization of the incident electron beam. If, for example, C 1 = C 1, we have the transverse polarization since the primary beam is totally polarized in the x direction (see (3.28)), while the cases C 1 -- 0 and C~ = 0 correspond to the total longitudinal parallel and antiparallel polarizations of the incident beam, respectively. From (3.30) the scattering cross section in the case of the transverse incident electron beam polarization are given by

d'do---~=(If+~(k-k')12+lf-~(k'

0 -k)lZ)c°s2 2 +

(If-/(

k'

-k)12-1f+l( k' -

k)12) sin2

0 (3.31)

and in the longitudinal case of the polarization they are

d--~dtr=lfl(k'-k)[2[ cOsz02 -

( kO~z'g/sin220]

(3.32a)

K. Stachulec / Spin polarized low energy electron diffraction

345

or

[

ddo" O=lf-~(k'-k)[

z cos 2 0 -

~0

• 2 ~0 sin

]

(3.32b)

for the parallel and antiparallel spin directions, respectively. We can see that the scattering cross section is evidently dependent on a state of the incident electron beam polarization, and for any state of the polarization it is expressed by means of the nonrelativistic amplitudes f +l(k' - k) and f_l(k' - k). The last amplitude can be calculated by means of the formula

~me f~0(k' - k) =

h2

2

( ~ exp(-i(k' - k)R~j~) exp ~j~

I(k'-_k)12B~] 2 /

oo

x ~] (1/B~)2k-~ I?T(B~ , So) k=0 ( 2 k + 1)!

(3.33)

which is the final result of the expression (3.23) after the integration over the variable r and insertion of the explicit form of Vx(r, So) given by (2.26). Due to the translation symmetry of the thin film the formula (3.33) can be written as follows:

f~(K) = Ftl ~ F~(K, So) e -½r2B" ,

(3.34)

~S

where K = ( k ' - k ) is the scattering vector which we decompose into components parallel and perpendicular to the surface of the film (K = (K~, KII)). The quantities we have introduced in the expression (3.34) are defined by Fll = ~] e-UqJ~

(3.34a)

J~

and

F~,(g, so)=

~--~ m e 2 h2 exp(-iK.c~v-ikp,)

Z (1/Bvs)Zk-1 ('T(Bvs So) ~=0 ( 2 k + l ) ! ' "

(3.34b)

Now one can see that the scattering amplitude including the spin is the sum of the partial scattering amplitudes of the monatomic layers parallel to the surface, each multiplied by its own factor exp(-½K2Bvs). This factor is different for each monatomic layer parallel to the surface, and it represents the Debye-Waller factor of the (vs)th layer of the film. We shall point out at this point that the temperature dependence of the scattering amplitude of the thin film is different from those of the bulk body because in the thin film case the scattering amplitude depends on temperature due to the Debye-Waller factors as well as due to an additional temperature dependence of the partial scattering amplitudes Fs~, whose Bus are dependent quantities. Eqs. (3.34), except their first Born approximation, are exact and together with (2.26) they denote the main results of the paper. The expressions for the scattering amplitude and for the scattering cross section are true in the L E E D as well as in the high energy electron diffraction ( H E E D ) cases. In the L E E D case, however, one can utilize the fact of low electron energy, which means that there the quantity ko/K in the formulae for the cross section can be replaced by 1.

346

K. Stachulec / Spin polarized low energy electron diffraction

4. Conclusions In this paper we have presented the theoretical model of the description of the spin polarized low energy electron diffraction by solid state sample with a free surface. The decomposition of the SPLEED amplitude for the electrons scattered by thin film into the component amplitudes of the given monatomic layers parallel to the surface is a natural consequence of the introduction of the dynamical scattering potential, which depends on temperature as well as on the position of the scattering atom in the film. The presence of the substrate on which the thin film is evaporated creates some boundary conditions for the thin film electrons as well as for the lattice dynamics of the film and these boundary conditions can be taken into calculation of the quantities B~ and (nJvs) by means of which the dynamical potential has been expressed. The last quantities can be calculated using the quantum field methods developed in thin film theories [5-8]. For the high temperature limit and simple cubic lattice case, excluding the spin, the scattering amplitude (3.34) can be written as

f(K) = FII ~ Fv(K ) e -~1¢28~ , v

where Fll = ~ e -ix/v and the component amplitude F~(K) is given by Jv

F.(K) =

W~me

hZ

2

e x p ( - i K l c , v)(/T(B~, (n~)),

while by 17-r(B~, (n~)) we have denoted the vth atom scattering potential excluding the exchange part (see (2.26)). In conclusion, we should like to draw attention to the fact that our approach to the SPLEED problem is based mainly on the behaviour of the dynamical surface scattering potential we have introduced and described in section 2. This potential in comparison with the existing theory of SPLEED (ref. 13 and references therein) removes the Coulomb singularities at the ion sites of the scattering sample in a natural way. It is therefore not necessary to replace the scattering potential of the ion-cores by the pseudopotentials as was done by Feder [13], and thus the computations of LEED [8] as well as SPLEED amplitudes are much simpler in our approach than those of Feder. For SPLEED, however, in our theory detailed calculations of the electron density distributions in thin films for different spin directions are necessary to compute the scattering intensity by means of the formulae (3.34). At present numerical computations of the problem are in progress and the results will be presented and compared with those of Feder in a supplementary paper.

Acknowledgement The author is much indebted to Professor L. Wojtczak for stimulating discussions and helpful criticism throughout the work.

References [1] J. Kessler, Polarized Electrons (Spinger-Verlag, Berlin, Heidelberg, New York, 1976). [2] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, New York, 1974).

K. Stachulec / Spin polarized low energy electron diffraction [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

J.C. Slater, The Self-consistent Field for Molecules and Solids (McGraw-Hill, New York, 1974) Vol. 4. A. Stachulec, Acta Phys. Hung. 51 (1981) 243. L. Valenta and L.Wojtczak, Czech. J. Phys. B30 (1980) 1025. L. Wojtczak, S. Romanowski and W. Stasiak, B. Mrygon, Czech. J. Phys. B31 (1981) 1024. K. Stachulec, Acta Phys. Hung. 54 (1983) 267. K. Stachulec, Acta Phys. Hung. 57 (1985) 55. F. Dworiankin, Kristalografia 10 (1965) 242. E. Clementi, Tables of Atomic Functions, IBM J. Res. and Develop. 9 (suppl.) (1965) pp. 2-70. I. S. Gradshteyn and I.M.Rizhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965). T.G. Strand and T. Tietz, Nuovo Cimento 41B (1966) 89. R. Feder, J. Phys. C14 (1981) 2049.

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