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arXiv:cond-mat/0305220v1 [cond-mat.mtrl-sci] 9 May 2003

X-ray resonant magnetic scattering from structurally and

magnetically rough interfaces in multilayered systems

I. Specular reflectivity

D. R. Lee,∗D. Haskel, Y. Choi,†J. C. Lang, S. A. Stepanov, and G. Srajer

Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439

S. K. Sinha

Department of Physics, University of California,

San Diego, La Jolla, CA 92093, and

Los Alamos National Laboratory, Los Alamos, NM 87545

(Dated: February 2, 2008)

Abstract

The theoretical formulation of x-ray resonant magnetic scattering from rough surfaces and in-

terfaces is given for specular reflectivity. A general expression is derived for both structurally and

magnetically rough interfaces in the distorted-wave Born approximation (DWBA) as the frame-

work of the theory. For this purpose, we have defined a “structural” and a “magnetic” interface

to represent the actual interfaces. A generalization of the well-known Nevot-Croce formula for

specular reflectivity is obtained for the case of a single rough magnetic interface using the self-

consistent method. Finally, the results are generalized to the case of multiple interfaces, as in the

case of thin films or multilayers. Theoretical calculations for each of the cases are illustrated with

numerical examples and compared with experimental results of magnetic reflectivity from a Gd/Fe

multilayer.

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I.INTRODUCTION

X-ray reflectivity and offspecular diffuse scattering methods have been widely applied

over the last decade to characterize the morphology of rough surfaces and interfaces, partic-

ularly with the availability of sources of ever-increasing brilliance for x-ray radiation. Similar

techniques using neutron beams have also become widespread, particularly for the study of

magnetic multilayers. In the case of x-rays, however, element-specific information regarding

the magnetic structure can be readily obtained by tuning the photon energy to that of an

L-edge (in the case of transition or rare-earth metals)1,2or of an M-edge (in the case of

actinides).3,4The resonant enhancement of the scattering by magnetic atoms at such ener-

gies can result in a large enough signal to be comparable to the dominant charge scattering.

Resonant x-ray scattering at the K-edges of transition metals5has also been used to obtain

information about the magnetic structure, although the enhancement is not as large. Reso-

nant magnetic scattering corresponds to the real part of the scattering amplitude, while the

(absorptive) imaginary part gives rise to x-ray magnetic circular dichroism (XMCD), which

has been used to obtain the values of spin and orbital moments in ferromagnetic materi-

als. Detailed descriptions of the formalism for the interaction of x-rays with magnetically

polarized atoms have been given in the literature,6–10from which a complete description of

magneto-optic phenomena in the x-ray region can be obtained and applied.

Several resonant x-ray specular reflectivity experiments have been performed to obtain

the magnetization within the layers of magnetic multilayers.2,11–14The analysis of these re-

sults has generally used recursive matrix techniques developed for magneto-optics in the

case of resonant x-ray reflectivity.15In general, roughness at the interfaces has been ignored

or taken into account in an ad-hoc manner. In principle, representing roughness in terms

of a graded magnetization at the interface and using slicing methods could enable one to

calculate the effect of magnetic roughness on specular reflectivity at the expense of con-

siderable computational effort. R¨ ohlsberger has developed a matrix formalism (originally

developed for nuclear resonant x-ray reflectivity) from which specular reflectivity incorpo-

rating roughness can be calculated.16It was not considered in his paper, however, that the

magnetic interfaces can have different roughnesses from the structural (chemical) ones. In

this paper, we define separately a structural and a magnetic interface to represent the actual

interfaces and present analytical formulae taking into account both interface roughnesses,

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which provide much faster computational method than the slicing methods and show good

agreement with established formulae for chemical interface roughness.

Methods were developed earlier to calculate analytically the specular component of the

charge scattering of x-rays by rough surfaces and interfaces using the Born approximation

(BA) and the distorted-wave Born approximation (DWBA).17,18The BA results were ex-

tended to magnetic interfaces in an earlier publication19and have already been applied

to interpreting x-ray resonant magnetic specular reflectivity measurements from magnetic

multilayers.14However, the BA or the kinematical approximation breaks down in the vicin-

ity of the critical angle and below, since it neglects the x-ray refraction. On the other hand,

the DWBA takes account of dynamical effects, such as multiple scattering and the x-ray re-

fraction, which become significant for smaller angles close to the critical angle and even for

greater angles at the resonant energies or with soft x-rays. We present here the generaliza-

tion of the DWBA to the case of resonant magnetic x-ray reflectivity from rough magnetic

surfaces or interfaces. The principal complication is, however, that we now have to deal

with a tensor (rather than scalar) scattering length, or equivalently an anisotropic refractive

index for x-rays.15This leads in general to two transmitted and two reflected waves at each

interface for arbitrary polarization, which complicates the DWBA formalism.

The plan of this paper is as follows. In Sec. II, we discuss a simple conceptual model

for a magnetic interface and its relationship to the chemical (i.e., structural) interface and

define the appropriate magnetic roughness parameters. In Sec. III, we discuss the (known)

scattering amplitudes for resonant x-ray scattering and their relationship to the dielectric

susceptibility to be used in the DWBA. In Sec. IV, we present the derivation of the scattering

in the DWBA for a single interface with both structural and magnetic roughnesses. In Secs.

V and VI, we derive the formulae for specular reflectivity from a magnetic interface using

the self-consistent method in the framework of the DWBA and discuss numerical results.

Finally, in Secs. VII-IX, we discuss the extension of the formalism to the case of the specular

reflectivity from magnetic multilayers and present some numerical results with experimental

data from a Gd/Fe multilayer. In the following paper,20we derive the formulae for the

diffuse (off-specular) scattering from magnetic interfaces in both the BA and the DWBA.

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II.MODEL FOR MAGNETIC INTERFACE

Consider an interface between a ferromagnetic medium and a nonmagnetic medium

(which could also be free space). Due to the roughness of this interface, the magnetic

moments near the interface will find themselves in anisotropy and exhange fields, which

fluctuate spatially (see Fig. 1).

This will produce disorder relative to the preferred ferromagnetic alignment within the

magnetic medium. A similar situation can arise at an interface between a ferromagnetic

medium (FM) and an antiferromagnetic medium (AFM), where there is a strong antiferro-

magnetic coupling between spins in the FM and the AFM. Random steps will then produce

frustration in the vicinity of the interface, resulting in random disordering of the magnetic

moments near the interface. Clearly in general correlation will exist between the height

fluctuations of the chemical interface and the fluctuations of the spins, but a quantitative

formalism to account for this in detail has not yet been developed. We make here the sim-

plifying assumption that the ferromagnetic moments near the interface (or at least their

components in the direction of the ferromagnetic moments deep within the FM layer, i.e.,

the direction of average magnetizationˆ M) are cut off at a mathematical interface, which we

call the magnetic interface and which may not coincide with the chemical interface, either

in its height fluctuations or over its average position, e.g., if a magnetic “dead layer” exists

between the two interfaces (see Fig. 1). The disorder near the interface is thus represented

by height fluctuations of this magnetic interface. The basis for this assumption, which is

admittedly crude, is that the short (i.e., atomic) length-scale fluctuations of the moments

away from the direction of the average magnetization give rise to diffuse scattering at fairly

large scattering wave vectors, whereas we are dealing here with scattering at a small wave

vector q, which represent the relatively slow variations of the average magnetization density.

The actual interface can be then considered as really composed of two interfaces, a chemi-

cal interface and a magnetic interface, each with their own average height, roughness, and

correlation length, and, importantly, in general possessing correlated height fluctuations.

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III.RESONANT MAGNETIC X-RAY SCATTERING AMPLITUDE

The amplitude for resonant magnetic scattering of x-rays has been derived by Hannon

et al.,6and a discussion of the general formalism may be found in the review by Hill and

McMorrow.9There are two cases of practical importance, namely dipole and quadrupole

resonances. We shall restrict ourselves here to the most commonly used dipole resonance,

which is related to the L-edges of transition metals and rare-earth atoms.The tensor

amplitude for scattering fαβfrom a magnetic atom is given by

?

αβ

e∗

fαfαβeiβ =

?

f0+3λ

8π(F11+ F1−1)

??

ˆ e∗

f·ˆ ei

?

− i3λ

3λ

8π(2F10− F11− F1−1)

8π(F11− F1−1)

?

ˆ e∗

f× ˆ ei

?

·ˆ M

+

?

ˆ e∗

f·ˆ M

??

ˆ ei·ˆ M

?

,(3.1)

where ˆ ei, ˆ ef are, respectively, the unit photon polarization vectors for the incident and

scattered waves,ˆ M is a unit vector in the direction of the magnetic moment of the atom,

λ is the x-ray photon wavelength, f0is the usual Thomson (charge) scattering amplitude

[f0 = −r0(Z + f′− if′′)], where r0 is the Thomson scattering length (e2/mc2), Z is the

atomic number, f′(< 0) and f′′(> 0) are the real and imaginary non-resonant dispersion

corrections. FLM is the resonant scattering amplitude, as defined in Ref. 6, and has the

resonant denominator Eres−E−iΓ/2, which provides the resonance when the photon energy

E is tuned to the resonant energy Eresclose to the absorption edges. The lifetime of the

resonance Γ is typically 1−10 eV, so that the necessary energy resolution is easily achivable

at synchrotron radiation beamlines. (We assumed that q, the wave-vector transfer, is small

enough here that the atomic form factor can be taken as unity.) Equation (3.1) has both

real and imaginary (i.e., absorptive) components. The latter gives rise to the well-known

phenomenon of x-ray magnetic circular or linear dichroism, whereas the real part gives rise

to the scattering. Equation (3.1) yields

fαβ= Aδαβ− iB

?

γ

ǫαβγMγ+ CMαMβ,(3.2)

where

A = f0+3λ

8π(F11+ F1−1),

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B =

3λ

8π(F11− F1−1),

3λ

8π(2F10− F11− F1−1),

C =

(3.3)

and α, β denote Cartesian components, and ǫαβγis the antisymmetric Levi-Civita symbol

(ǫxyz = ǫyzx = ǫzxy = 1, ǫxzy = ǫyxz = ǫzyx = −1, all other ǫαβγ = 0). The dielectric

susceptibility of a resonant magnetic medium is given by

χresonant

αβ

(r) =4π

k2

0

nm(r)fαβ(r),(3.4)

where k0= 2π/λ, nm(r) is the local number density of resonant magnetic atoms, and the

variation of fαβ(r) with r reflects the possible positional dependence of the direction of

magnetization M. The total dielectric susceptibility is given by

χαβ(r) =

4π

k2

0

??

−ρ0(r)r0+ Anm(r)

?

δαβ

− iBnm(r)

?

γ

ǫαβγMγ(r) + Cnm(r)Mα(r)Mβ(r)

?

, (3.5)

where ρ0(r) represents the electron number density arising from all the other nonresonant

atoms in the medium modified by their anomalous dispersion corrections when necessary.

Using the constitutive relationship between the local dielectric constant tensor ǫαβ(r) and

χαβ(r),

ǫαβ(r) = δαβ+ χαβ(r).(3.6)

We note that the magnetization gives the dielectric tensor the same symmetry as in conven-

tional magneto-optic theory, namely an antisymmetric component linear in the magnetiza-

tion.

IV. THE DISTORTED-WAVE BORN APPROXIMATION FOR A SINGLE MAG-

NETIC INTERFACE

The results for specular reflectivity in the Born approximation (BA) have been derived

in Ref. 19 and will be also summarized briefly in connection with the cross section in

the following paper.20Here we discuss the scattering in terms of the distorted-wave Born

approximation (DWBA). While this is more complicated algebraically, it provides a better

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description than the simple kinematical approximation or BA in the vicinity of regions where

total reflection or Bragg scattering occurs. This treatment is a generalization of that used

in Ref. 17 for charge scattering. The wave equation for electromagnetic waves propagating

in an anisotropic medium with a dielectric susceptibility tensor given by Eq. (3.5) may be

written as

?

β

?

(∇2+ k2

0)δαβ− ∇α∇β+ k2

0χαβ

?

Eβ(r) = 0, (α,β = x,y,z),(4.1)

where E(r) is the electric field vector.

Consider a wave incident, as in Fig. 1 with wave vector kiin the (x,z) plane (ki,y= 0) and

polarization µ (µ = σ or π), from a nonmagnetic (isotropic) medium for which χαβ= χ0δαβ

onto a smooth interface at z = 0 with a magnetic medium, for which χαβ is constant for

z < 0.

Let us write for z < 0

χαβ= χ1δαβ+ χ(2)

αβ, (4.2)

where the term χ(2)

αβis the part that specifically depends on the magnetization M, as defined

in Eq. (3.5). The incident wave (chosen for convenience with unit amplitude) may be written

as

Ei

µ(r) = ˆ eµeiki·r. (4.3)

This incident wave will in general give rise to two specularly reflected waves (where the

index µ refers to σ or π polarization) and two transmitted (refracted) waves in the magnetic

medium. The complete solution for the electric field in the case of the smooth magnetic

interface is then given by

E(ki,µ)(r) = ˆ eµeiki·r+

?

ν=σ,π

R(0)

νµ(ki)ˆ eνeikr

i·r, z > 0,

=

?

j=1,2

T(0)

jµ(ki)ˆ ejeikt

i(j)·r, z < 0,(4.4)

where kr

iis the specularly reflected wave vector in the nonmagnetic medium, ν denotes

the polarization of the appropriate reflected component, the index j(= 1,2) defines the

component of the transmitted wave in the magnetic resonant medium with polarization ˆ ej

(ˆ ej=1,2= ˆ e(1)and ˆ e(2), respectively, as defined in Appendix A), and kt

i(j) the appropriate

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wave vector for that transmitted wave. The polarization vectors ˆ e may be real or complex

allowing for linear or elliptically polarized waves.We denote such states in Eq.(4.4)

quantum-mechanically by |ki,µ >.

R(0)

νµand T(0)

jµ denote the appropriate reflection and transmission coefficients for the

smooth surface and are expressed in terms of 2×2 matrices using the polarization bases for

the incident and reflected (or transmitted) waves. The polarization basis is given by (ˆ eσ,

ˆ eπ), as shown in Fig. 1, for the waves in the nonmagnetic medium and (ˆ e(1), ˆ e(2)), as defined

in Appendix A, for those in the magnetic resonant medium, respectively. The convention

in which the polarization state of the reflected (or transmitted) wave precedes that of the

incident wave is used for the subscripts in R(0)

νµand T(0)

jµ, and the Greek and Roman letters

are used for the polarization states in the nonmagnetic and magnetic medium, respectively.

The explicit expressions of R(0)

νµand T(0)

jµfor small angles of incidence and small amplitudes of

the dielectric susceptibility and for special directions of the polarization and magnetization

(i.e., M ? ˆ x as shown in Fig. 1) are given in Appendix A.

We should mention, however, that these specific conditions considered in Appendix A

(and also in all other appendices) are reasonably satisfied for hard- and medium-energy x-

rays and also for soft x-rays around transition-metal L-edges with small angles (i.e., when

θ2

i≪ 1 for the incidence angle θi). We should also mention that, even when M is not

parallel to the ˆ x-axis in Fig. 1, the expressions derived in the appendices can be still

applied by considering only the x-component of the magnetization vector M. This is because

the y- and z-components of M contribute negligibly to the scattering in comparison with

with the dominant factor B = (3λ/8π)(F11− F1−1) in Eq. (3.2) at small angles15when

|F11− F1−1| ≫ |2F10− F11− F1−1|, which is generally satisfied for transition-metal and

rare-earth L-edges.8

We note that the continuity of the fields parallel to the interface requires that

(ki)?= (kr

i)?=

?

kt

i(j)

?

?, (4.5)

where ()?denotes the vector component parallel to the interface.

We now discuss the structurally and magnetically rough interface. For this purpose we

shall assume that the average height (along z) of the structural and magnetic interfaces is

the same, i.e., we ignore the presence of a magnetic dead layer. This may be treated within

the DWBA as simply another nonmagnetic layer and thus discussed within the formalism

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for treating multilayers as discussed in Section VII. We can write

χαβ(r) = χ(0)

αβ(r) + ∆c

αβ(r) + ∆m

αβ(r),(4.6)

where

χ(0)

αβ(r) = χ0δαβ, z > 0

= χ1δαβ+ χ(2)

αβ, z < 0,(4.7)

∆c

αβ(r) = (χ1− χ0)δαβ, for 0 < z < δzc(x,y) if δzc(x,y) > 0

= −(χ1− χ0)δαβ, for δzc(x,y) < z < 0 if δzc(x,y) < 0

= 0 elsewhere, (4.8)

and

∆m

αβ(r) = χ(2)

αβ, for 0 < z < δzm(x,y) if δzm(x,y) > 0

= −χ(2)

= 0

αβ, for δzm(x,y) < z < 0 if δzm(x,y) < 0

elsewhere, (4.9)

δzc(x,y) and δzm(x,y) define the structural (chemical) and magnetic interfaces, respectively.

We may also define the time-reversed function corresponding to a wave incident on the

interface with vector (−kf) and polarization ν as

ET

(−kf,ν)(r) = ˆ eνeik∗

f·r+

?

λ=σ,π

R(0)∗

λν(−kf)ˆ eλeikr∗

f·r, z > 0

=

?

j=1,2

T(0)∗

jν (−kf)ˆ ejeikt∗

f(j)·r, z < 0,(4.10)

where (−kr

is the wave vector of one of the two transmitted waves in the medium emanating from

f) is the wave vector of the wave specularly reflected from (−kf), and

?

−kt

f(j)

?

(−kf) incident on the surface, as shown in Fig. 2. Note that, for consistency with the

conventions used in Eq. (4.4), the polarization vectors in Eq. (4.10) are defined in the

ordinary coordinate system where their phases are considered along the left-to-right direction

in Fig. 1. Otherwise, the polarization vectors in Eq. (4.10) should be replaced by their

complex conjugates.

We have also the conditions

(kf)?= (kr

f)?=

?

kt

f(j)

?

?.(4.11)

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The DWBA then yields the differential cross section for scattering by the rough interface

from (ki,µ) to (kf,ν) as

dσ

dΩ=

1

16π2

?

|Tfi|2?

,(4.12)

where Tfi=< kf,ν|T |ki,µ > is the scattering matrix element, and

denotes a statistical averaging over random fluctuations at the interface. Following Ref. 17,

?

...

?

in Eq. (4.12)

we split the cross section into two parts:

dσ

dΩ=

1

16π2

???

?

Tfi????

2+

1

16π2

??

|Tfi|2?

−

???

?

Tfi????

2?

. (4.13)

The first term in Eq. (4.13) represents the coherent (specular) part of the scattering, which

corresponds to a statistical averaging of the scattering amplitude, and the second term

corresponds to the incoherent (diffuse) scattering. In this paper, we shall deal with the first

term only, while the diffuse scattering will be addressed in the following paper.20

The DWBA consists of approximating the scattering matrix element by the expression

< kf,ν|T |ki,µ > = k2

0< −kT

0< −kT

f,ν|χ(0)|Ei

f,ν|∆c|ki,µ > +k2

µ(r) >

+ k2

0< −kT

f,ν|∆m|ki,µ > .(4.14)

Here |Ei

in Eq. (4.10), and the matrix element involves dot products of the tensor operators χ(0), ∆c,

µ(r) > denotes the “pure” incoming wave in Eq. (4.3), |−kT

f,ν > denotes the state

and ∆mwith the vector fields < −kT

with a smooth interface, ∆cand ∆mare perturbations on χ(0)due to interface roughnesses.

f,ν| and < ki,µ|. While χ(0)represents an ideal system

For the smooth surface, only the first tensor is nonvanishing, and, following Ref. 17, we

can show from Eqs. (4.3) and (4.10) that

k2

0< −kT

f,ν|χ(0)|Ei

µ(r) > = iAk2

×

0δkixkfxδkiykfy

?

?0

j

T(0)

jν(−kf)

?

αβ

e∗

jα(χ1δαβ+ χ(2)

αβ)eµβ

×

= 2iAkizR(0)

−∞dze−i(kt

fz(j)−kiz)z,

νµ(ki)δkixkfxδkiykfy,(4.15)

where A is the illuminated surface area, and R(0)

smooth surface, as defined in Eq. (4.4). The details of Eq. (4.15) are presented in Appendix

νµ(ki) is the reflection coefficient for the

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B. By comparison with Eq. (4.15) for the smooth surface, the scattering matrix element for

the rough surface in Eq. (4.14) can be analogously defined by

< kf,ν|T |ki,µ >= 2iAkizRνµ(ki)δkixkfxδkiykfy,(4.16)

where Rνµ(ki) denotes the reflection coefficient for the rough surface.

On the other hand, for the reverse case where a wave is incident from a resonant magnetic

medium to a nonmagnetic (isotropic) medium, similarly to Eq. (4.15), the scattering matrix

element for the smooth surface can be shown to be

k2

0< −kT

f,j′|χ(0)|ki,j >= 4iAkiz(j)R(0)

j′j(ki)δkixkfxδkiykfy, (4.17)

where the incoming wave from the resonant magnetic medium |ki,j > is used instead of

the “pure” incoming wave from the vacuum Ei

µ(r) in Eq. (4.3). The use of Eqs. (4.15)

and (4.17) in Eqs. (4.14) and (4.12) in the case of the smooth surface and the derivation

of the corresponding reflectivity in the usual manner, as discussed in Ref. 17, shows that

Eqs. (4.15) and (4.17) must be identically true. Similarly to Eqs. (4.15) and (4.16), the

scattering matrix element for the rough surface between reversed layers can be also defined

by analogy from Eq. (4.17) as

< kf,j′|T |ki,j >= 4iAkiz(j)Rj′j(ki)δkixkfxδkiykfy,(4.18)

where Rj′j(ki) denotes the reflection coefficient for the rough surface between reversed layers.

V.REFLECTION AND TRANSMISSION COEFFICIENTS USING THE SELF-

CONSISTENT METHOD

To calculate specular reflectivity, we make an approximation in the spirit of Nevot and

Croce.21To evaluate the matrix elements in Eq. (4.14) involving ∆c

αβand ∆m

αβ, we assume

for E(ki,µ) in Eq. (4.4) the functional form for z > 0 analytically continued for z < 0,

while for the time-reversed state ET(−kf,ν) in Eq. (4.10) the functional form for z < 0

analytically continued to z > 0. Then, bearing in mind that for specular reflectivity kf= kr

i

and using Eq. (4.5), we obtain for the statistically averaged amplitude

?

Tfi?

:

?

k2

0< −kT

f,ν|∆c,m|ki,µ >

?

= iAk2

0

?

j=1,2

T(0)

jν(−kf)

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×

??

αβ

e∗

jα∆c,m

q1z(j)

αβeµβ

??

e∗

e−iq1z(j)δzc,m(x,y)?

jα∆c,m

q2z(j)

− 1

?

+

?

λ=σ,π

R(0)

λµ(ki)

?

αβ

αβeλβ

??

e−iq2z(j)δzc,m(x,y)?

− 1

??

, (5.1)

where

q1z(j) = kt

fz(j) − kiz,q2z(j) = kt

fz(j) − kr

iz, (5.2)

and ∆c,m

αβis the value defined for 0 < z < δzc,min Eqs. (4.8) and (4.9). From Eqs. (4.15)-

(4.16) and (5.1), we see that, at the specular condition, we can write Eq. (4.14) as

Rνµ= R(0)

νµ+ Uνµ+

?

λ

VνλR(0)

λµ, (5.3)

where

Uνµ =

?

j=1,2

T(0)

jν(−kf)

2kiz

k2

0

q1z(j)

?

(χ1− χ0)

?

α

e∗

jαeµα[e−1

2q2

1z(j)σ2

c− 1]

+

?

αβ

e∗

jαχ(2)

αβeµβ[e−1

2q2

1z(j)σ2

m− 1]

?

, (5.4)

and replacing q1z, eµin Uνµby q2z, eλproduces Vµλ. Here we made the customary Gaussian

approximation for the height fluctuations δzc,m(x,y), and σc, σmare the root-mean-squared

structural and magnetic roughnesses, respectively. Note that the correlation term Uνµdue

to the roughness in the reflection coefficient contains only independent contributions of

chemical and magnetic roughnesses expressed via σcand σm, respectively. According to Eq.

(4.13), the diffuse scattering must contain the cross-correlation component due to the term

|Tfi|2?

A better approximation than Eq. (5.3) may be obtained by using the rough-interface

?

.

reflection coefficient Rνµinstead of the smooth-interface R(0)

νµin the wave functions of Eqs.

(4.4) and (4.10), thus getting a self-consistent matrix equation in terms of the 2×2 matrices,

R, U, V. This leads to

R = R(0)+ U + VR,(5.5)

whose solution is

R = (1 − V)−1(R(0)+ U).(5.6)

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Similarly, for the reverse interface between upper resonant magnetic and lower nonmagnetic

layers, we can have the same solution as Eq. (5.6) from Eqs. (4.17) and (4.18). The explicit

expressions of U, V, R(0)matrices in Eq. (5.6) for both cases are given in Appendix C.

For nonmagnetic interfaces, the matrices are all diagonal(σ and π polarizations are de-

coupled), and it has been shown that Eq. (5.6) leads to the familiar Nevot-Croce form21for

the reflection coefficient, i.e.,

R = R(0)e−2|kz||kt

z|σ2

c.(5.7)

The derivation of this is shown in Appendix D. For the magnetic interface, this simplified

form for the reflection coefficient does not have any analogue. Nevertheless, at sufficiently

large values of qz, the reflectivity takes the familiar Gaussian form R(0)e−q2

zσ2

eff. However, σ2

eff

does not always take the form predicted by the simple kinematical theory [i.e., σ2

cfor σ → σ

reflectivity, σ2

mfor σ → π reflectivity, and1

polarized x-rays] as we shall see in the numerical example shown below, which provides a

2(σ2

c+ σ2

m) for (I+− I−) in the case of circularly

counter-illustration of the rule that, at large qz, the DWBA becomes identical to the Born

approximation or kinematical limit.

For circularly polarized incident x-rays with ˆ e±(?ki) =

tion amplitudes for σ- and π-polarization are given by

?

ˆ eσ(?ki) ± iˆ eπ(?ki)

?

/√2, the reflec-

Rσ

Rπ

= R

1

√2

±

i

√2

, (5.8)

where R is the 2×2 matrix reflection coefficient in Eq. (5.6). The reflected intensities without

polarization analysis for the outgoing beam, I =

?

|Rσ|2+ |Rπ|2, can be then evaluated for

the opposite helicities of incident beams as

I+− I−= 2 Im[R11R∗

12+ R21R∗

22], (5.9)

where Rijis the ij-element of the 2 × 2 matrix R.

Since Parratt’s recursive formula for multiple interfaces includes only reflection coeffi-

cients, its extension to the rough interface case does not need the transmission coefficient

to account for interface roughness. On the other hand, in our case where the fields are not

scalars, the transmission coefficients are requisite to calculate recursive 2×2 matrix formu-

lae for multiple magnetic interfaces, which will be discussed in Sec. VII. For completeness,

13

Page 14

therefore, let us now calculate the transmission coefficient Tjµfrom a rough interface. In the

spirit of Ref. 22, we assume for E(ki,µ) and ET(−kf,j) the functional forms analytically

continued both for z > 0 and for z < 0 as follows:

E(ki,µ) =

?

?

j′=1,2

T(0)

j′µ(ki)ˆ ej′eikt

i(j′)·r, (5.10)

ET(−kf,j) =

ν=σ,π

T(0)∗

νj (−kf)ˆ eνe−ik∗

f·r, (5.11)

where T(0)∗

νj (−kf) in Eq.

netic(anisotropic) medium “to” a nonmagnetic (isotropic) one, whose explicit form is given in

(5.11) denotes the transmission coefficient “from” a mag-

Appendix A. For the smooth surface, the scattering matrix element between the eigenstates

| − kT

f,j > and |ki,µ > can be then written as

k2

0< −kT

f,j|χ(0)|ki,µ > = iAk2

0δkixkfxδkiykfy

×

?

?0

ν

T(0)

νj(−kf)

?

j′

T(0)

j′µ(ki)

?

αβ

e∗

να(χ1δαβ+ χ(2)

αβ)ej′β

×

= 4iAkt

−∞dze−i(−kfz−kt

iz(j)T(0)

iz(j′))z,

jµ(ki)δkixkfxδkiykfy,(5.12)

where T(0)

jµ(ki) is the transmission coefficient for the smooth surface, as defined in Eq. (4.4).

The details of Eqs. (5.12) are given in Appendix B.

In comparison with Eq. (5.12) for the smooth surface, the scattering matrix element for

the rough surface, as shown in Eq. (4.14), can be analogously defined by

< kf,j|T |ki,µ >= 4iAkt

iz(j)Tjµ(ki)δkixkfxδkiykfy,(5.13)

where Tjµ(ki) denotes the transmission coefficient for the rough surface.

For the statistically averaged amplitude

?

Tfi?

T(0)

, we obtain

?

k2

0< −kT

f,j|∆c,m|ki,µ >

?

= iAk2

0

?

ν

νj(−kf)

×

?

j′

T(0)

j′µ(ki)

?

αβ

e∗

να∆c,m

q3z(j′)

αβej′β

??

e−iq3z(j′)δzc,m(x,y)?

− 1

?

, (5.14)

and

q3z(j′) = −kfz− kt

iz(j′).(5.15)

14

Page 15

From Eqs. (5.12)-(5.13) and (5.14), we see that we can write the scattering matrix

element in the DWBA, as shown in Eq. (4.14), as

Tjµ= T(0)

jµ+

?

j′=1,2

V′

jj′T(0)

j′µ,(5.16)

where

V′

jj′ =

?

ν

T(0)

4kt

νj(−kf)

iz(j)

k2

0

q3z(j′)

?

(χ1− χ0)

?

α

e∗

ναej′α

?

e−1

2q2

3z(j′)σ2

c− 1

?

+

?

αβ

e∗

ναχ(2)

αβej′β

?

e−1

2q2

3z(j′)σ2

m− 1

??

. (5.17)

In the same way as we did for the reflection coefficient, using the rough-interface trans-

mission coefficient Tjµinstead of the smooth-interface T(0)

jµin the right side of Eq. (5.16),

thus getting a self-consistent matrix equation in terms of the 2 × 2 matrices, T, V′, gives

T = T(0)+ V′T,(5.18)

whose solution is

T = (1 − V′)−1T(0). (5.19)

Similarly, for the reverse interface between upper resonant magnetic and lower nonmagnetic

layers, we can also have the same solution as Eq. (5.19). The explicit expressions of V′and

T(0)matrices in Eq. (5.19) for both cases are given in Appendix C.

For nonmagnetic interfaces, it is shown in Appendix D that Eq. (5.19) reduces to

T = T(0)e

1

2(|kz|−|kt

z|)

2σ2

c,(5.20)

which has been found by Vidal and Vincent.23

VI.NUMERICAL EXAMPLES FOR A SINGLE MAGNETIC SURFACE

We now illustrate numerical examples of the above formulae calculated for a Gd surface

with varying degrees of structural and magnetic roughness. We have considered only the

case where the magnetization vector is aligned along the sample surface in the scattering

plane in order to enhance the magnetic effect.

Figure 3 shows the x-ray resonant magnetic reflectivities calculated at the Gd L3-edge

(7243 eV) from Gd surfaces with different interfacial widths for structural (σc) and magnetic

15