OPTICAL 2ND-HARMONIC GENERATION FROM MAGNETIZED SURFACES

(1)

Optical

second-harmonic

generation

from

magnetized

surfaces

Ru-Pin Pan

Department ofElectrophysics, National Chiao Tung Uniuersity, Hsinchu, Taiwan 30049,Republic ofChina

H.

D.

Wei

Institute ofElectronics, National Chiao Tung Uniuersity, Hsinchu, Taiwan 30049,Republic

of

China

Y.

R.

Shen

Department

of

Physics, University ofCalifornia, Berkeley, California 94720 (Received 7July 1988)

We propose optical second-harmonic generation as ameans to probe surface magnetization. It is shown that surface magnetization can induce a number ofnonlinear susceptibility elements that

would vanish otherwise. They are presented for the (001), (110),and (111)surfaces ofafcc cen-trosymmetric crystal. An order-of-magnitude estimate, using the microscopic expression of the nonlinear susceptibility, suggests that these induced elements are detectable by optical second-harmonic generation with appropriate polarization combinations. The second-harmonic signals

from magnetized and nonmagnetized surfaces should exhibit characteristically different rotational anisotropy.

I.

INTRODUCTION

The controversy over the observation

of

magnetically

"dead"

_layers _by Liebermann et

al.

' almost two decades ago has sparked continuting intensive studies in the field

of

surface magnetism. ' From the basic research point

of

view, the fact that the surface and the bulk

of

the same material can have very different magnetic properties is certainly most interesting and intriguing. '

"

_Improved

understanding

of

the role

of

surfaces in magnetic phase transitions will also help shed light on other physical phe-nomena such as surface melting. From the application point

of

view, the fact that magnetic memory devices could be affected by surface magnetization naturally calls

for a better understanding

of

surface magnetism. The

surface magnetic properties

of

transition metals are

also-of

pivotal importance for catalysis and metallurgy.

Experimental studies in this field have, however, been impeded by the limited number

of

analytical tools avail-able. Techniques such as electron-capture spectroscopy,

inverse photoelectron spectroscopy, Hall-effect measure-ment, ' and angle-resolved photoelectron spectroscopy'

have been used. Most

of

these methods require placing

the probe along with the sample in an environment

of

ul-trahigh vacuum. Recently, optical second-harmonic gen-eration (SHG) has been proven to be a versatile probe for

surface studies. '

lt

has the advantages

of

being highly surface sensitive, capable

of

remote sensing and in situ measurement, and applicable to any interface accessible by 1ight. Gne may question whether the technique can also be used to probe surface magnetization. Recently,

Pan and Shen drew attention to this possibility by show-ing that the nonlinear optical susceptibility tensor for SHG for the (001)surface

of

a cubic crystal possesses a group

of

nonvanishing elements induced by the presence

of

afinite magnetization,

M,

parallel to the (100) axis.'

In this paper, we show in more detail the experimental feasibility

of

probing magnetized surface by optical

second-harmonic generation. The nonlinear optical sus-ceptibility tensors for SHG from magnetized (100),

(110),

and

(111)

surfaces

of

a fcc centrosymmetric crystal are derived and tabulated in

Sec.

II.

The input-output

polar-ization combinations needed for detecting the nonlinear susceptibility elements induced by surface magnetization

are suggested in

Sec.

III.

The SH signals from magnet-ized and nonmagnetmagnet-ized surfaces can exhibit different

ro-tational symmetries. Section IV shows a first-order

mi-croscopic expression

of

the magnetization-induced

sur-face nonlinear susceptibility, taking into account

spin-orbit interaction

of

the conduction electrons. An

order-of-magnitude estimate ispresented, using nickel as an

ex-ample.

II.

SYMMETRY CONSIDERATIONS

In a centrosymmetric medium, the electric dipole

con-tribution to the second-order optical nonlinearity is iden-tically zero. At the surface, the inversion symmetry is broken, resulting in the high surface sensitivity

of SHG.

The magnetization

of

a material will not break the inver-sion symmetry

of

the bulk material, but can lower the surface symmetry, and hence modify the form

of

the non-linear susceptibility tensors for surface

SHG.

The surface nonlinear optical polarization at

2'

can be written as

P,

(2co)

=y'

'(M):E(co)E(to),

where the surface nonlinear susceptibility third-rank

ten-sor g' ' _is _{a function}

_of

_{the surface magnetization} _M _in

general and E(co)is the fundamental field. The symmetry

of

g'

'(M)

is dictated by the symmetry

of

the particular

surface under consideration. The nonzero elements

of

(2)

1230 RU-PIN PAN, H.D.WEI, AND

Y.

R.

SHEN 39 TABLE

I.

Independent nonvanishing elements ofg', '_for

(001), (110),and (111)surfaces offcc crys-tals without surface magnetization. (Surface is inthe x-y plane.)

Surfaces (001) (110) (111) Symmetry classes 4m mm2

3'

Independent nonvanishing elements

xzx =xxz=yzy =yyz, zxx =zyy, zzz

xzx =xxz, yzy =yyz, zxx, zyy, zzz

xxx

=

—

xyy

= —

yxy

= —

yyx, xzx

=xxz

=yzy =yyz, zxx =zyy, zzz

X'

'(M)

can be obtained from invariance

of

X'

'(M)

under symmetry operations:

(2)

_J~=T,

—

(2)

,-.T)).

~kg;)k,

i,

z,

k=x,

y, z (2) where

T

is the transformation matrix for each symmetry operation, and summation over repeated indices is im-plied. The time-reversal properties

of

X'

_'(M),

_neglecting dissipation, requires that the real part

of

X'

'(M)

is an even function

of

M, while the imaginary part

of

X'

'(M)

is an odd function

of M.

The latter group, nonvanishing only in the presence

of

afinite M, can be useful for prob-ing surface magnetization.

We consider the (100),

(110),

and

(111)

surfaces

of fcc

centrosymmetric crystals. In the absence

of

surface mag-netization, the (001)and (110)faces are

of

4mm and mm2 symmetries, respectively, while the

(111)

surface is

of

class 3m when more than two monolayers

of

the surface

are considered. Nonvanishing elements

of

X'

'(0)

for

these symmetries can be found in numerous works, e.g., the recent paper by Guyot-Sionnest et al.' These are reproduced in Table

I

for later comparison. With surface magnetization, the symmetries are changed. We summa-rize our derivation for the non vanishing surface-susceptibility tensor elements in the following. The

coor-dinate system chosen forall cases isto have the surface in the x-y plane and the surface normal along

z.

First we consider a magnetized (001) surface: With M

parallel to the

[100]

direction (x),the fourfold symmetry

of

the surface about the surface normal is broken. The

two independent symmetry operations are (i) reflection about the x-z mirror plane,

x

~x,

y

~

—

y, and

M~

—

M,

and (ii) reflection about the y-z mirror plane,

x

~

—

x,

y~y,

and

M~M.

Using Eq. (2), one can show that there are ten independent nonvanishing elements:

(2) (2) (2)

—

(2) (2) (2) (2)

(Xxxz Xxzx~ _Xyyz _Xyzy~ Xzxx~ _Xzyy~ _Xzzz Xyl,

'),

X(y)=X(2y)) are odd in

M. If

the surface

magnetiza-tion M is parallel to the

[110]

direction, all

of

the 27 ele-ments are nonzero, ten

of

which are independent. Actu-ally, the latter can be obtained from the former case

of

M~~[100] by a coordinate transformation with x'~~x+y

and y'~~x

—

y.

If

the surface magnetization, M, is parallel

to the surface normal, i.e., the

[001]

direction, the

four-fold symmetry

of

the surface is preserved. The symmetry operations are (x

~x,

y

~

—

y, and M

~

—

M) and

(x

~

—

x,

y

~y,

and M

~

—

M)

for reflections about the x-zand y-z planes, respectively, yielding five independent nonvanishing elements. Three

of

them are even in M and the others odd in

M.

Table

II

lists all the nonvanishing susceptibility elements forthe above cases.

Similarly, we can find the nonvanishing

_g

' elements for a magnetized (110)surface as shown in Table

III.

We

choose the x and _y axes to be along the

[110]

and

[001]

directions, respectively.

For

M~~[001]~~y, there are ten independent nonvanishing elements; five

of

them are even and the other five odd in

M. For

the case

of

M~~[110]~~z, there are eight independent nonzero elements; five

of

them are even and the other three odd in

M.

Note that the symmetry

of

X'

'(M)

for the (110)surface is identical

tothat forthe (001)surface.

Finally, our results for a magnetized

(111)

surface are given in Table IV. The

[21

1]

and

[Ol1]

directions are

TABLE

II.

Independent nonvanishing elements of g,' _'(M)

for the (001) surface offcccrystals with

surface magnetization M parallel to the [100],[110],and [001]directions. (x is along the [001]direction

and the surface is in the x-y plane.)

Direction of M

Even parity Odd parity

[100] xzx =xyz, yzy =yyz,

zxx, zyy, zzz

xyx =xxy, yxx, yyy, yzz, zyz =zzy

[110] xyz =xzy =yzx =yxz,

xzx

=xxz

=yzy =yyz, zxx=zyy, zzz, zxy =zyx

xxx

= —

yyy, xyy

= —

yxx,

xzz

—

yzz,

xxy =xyx

= —

yxy

=

—

yyx, zxz =zzx

= —

zyz

=

—

zzy

[001] xzx

=xxz

=yzy =yyz,

zxx =zyy, zzz

xyz =xzy =yzx =yxz,

(3)

TABLE

III.

Independent nonvanishing elements _{of g,}' _'(M)

for the (110)surface offcccrystals with surface magnetization M parallel to [001]and [110]directions. (y is along the [001] direction and the surface is in the x-y plane.)

Direction of M

Independent nonvanishing elements Even parity Odd parity

C Y' =X' [001] xzx =xxz, yzy =yyz, zxx, zyy, zzz xxx, xyy, xzz, yxy =yyx, zxz=zzx 2 uJ

[110] xzx =xxz, yzy =yyz, xyz =xzy, yzx =yxz, Medium Z

zxx, zyy, zzz zxy =zyx

Medium

Z=O

chosen to be the xand _y axes. The cases for

_M~~[211]

~x

and M~~[111]~~zhave been studied. Eighteen and seven independent nonvanishing elements are found, respective-ly. The parities

of

these elements with respect to M are indicated in the table.

Ep

III.

SECOND-HARMONIC GENERATION

ASA PROBE FORSURFACE MAGNETISM Y'

Y

AI

We have shown that certain surface nonlinear suscepti-bility tensor elements are nonvanishing only in the pres-ence

of

a finite

M.

With appropriate combinations

of

in-put and outin-put beam polarizations, different elements

of

y

(M)

can be selectively measured by surface

SHG.

In the following discussion, we shall use the experimental geometry shown in

Fig. 1.

The interface

of

medium

I

(air

or vacuum) and medium

II

(which can be magnetized) is in the x-y plane, with the surface normal being the z axis. The laser beam is incident at an angle 0 with respect to the z axis, whereas L9' is the refraction angle in medium

II.

The x' and y' axes in the x-y plane are chosen to be parallel and perpendicular to the plane

of

incidence, re-spectively, and Pisthe angle between x and

x'.

The nota-tion (s,p) denotes that the incident fundamental beam is p polarized while the SH output is s polarized, and so on. The SH signal reflected from a centrosymmetric medium actually consists

of

not only a surface contribution but also a bulk contribution. The latter, however, tends to be weaker than the former in the case

of

metals. ' The

A A

Z, Z'

FIG.

1. (a) Geometry of second-harmonic generation in

reflection from an interface between two media. (b)Beam coor-dinates x',y', and z' relative to crystal coordinates x,y, and z.

effective surface nonlinear polarization can be written as

P,

tr=

P,

(M)

+

P„,

where

P,

(M),

defined in

Eq.

(1),is the surface contribu-tion to

P,

~and

P„

is the equivalent surface polarization due to the bulk quadrupole. '

'

We shall assume that only the surface layer is magnetized.

A. (001)surface

For

a normally incident beam with the (s,s) polariza-tion combinapolariza-tion, the reflected SH signal vanishes for

TABLE IV. Independent nonvanishing elements of

_g,

' _'(M) _{for the} ₍₁₁₁₎_surface _of_fcc_crystals _with

surface magnetization Mparallel to the [211],and [111]directions. (xis along the [211]direction and

the surface is in the x-y plane.)

Direction of M

[2

11]

xxx,xyy, xzz,

xzx =xxz, yzy =yyz,

yxy =yyx, zxx, zyy,

zzz, zxz

=

zzx

xyx =xzy, xxy =xyx,

yxx, yyy, yzz,

xzx =yxz, zyz =zzy,zxy =zyx

Even parity Odd parity

xxx

= —

xyy

=

—

yxy

= —

yyx, xzx

=xxz

=yyz =yzy, zxx, zyy, zzz

xyz =xzy

= —

yzx

= —

yxz, xxy =xyx =yxx

= —

yyy

(4)

1232 RU-PIN PAN, H.D.WEI, AND Y.

R.

SHEN 39

P

(M)

yyy

(M)E

E

Applying Eq.(2),we get

P,

,

(M)

=

_{[2y'~)

(M)+y'

'

(M)]sin

_PcosP

+y'

'(M)cos

_{P]E E}

~

C&cosp+

C2cos(3$),

(4) M

=0.

In the presence

of

amagnetized surface, M~~

[100]

or

[110],

the

y'

component

of

the surface nonlinear polar-ization is

where k (co) is the

x'

component

of

the wave vector

of

the pump field. _g is a phenomenological constant describing the anisotropic contribution to P&

.

'

For

a

normally incident s-polarized laser beam,

k„,

(co)

=0,

the reflected SH signal is generated by

P,

only.

If

the in-cidence angle is nonzero, the contribution

of

the surface magnetization to the nonlinear polarization can still be identified by inspecting the P dependence

of

the reflected SH signal.

With M~~[110],one can show that, for the (s,p) polar-ization combination,

where

C,

and Cz are linear combinations

of

the nonlinear susceptibility tensor elements and the pump fields.

It

is seen that

P,

~ has two components with onefold and

threefold rotational symmetries, respectively, about the surface normal ~ The equivalent surface nonlinear

polar-ization due tothe bulk quadrupole for an incident angle 0 1S]8

and

P,

,

=

[[yy,

,

'(M)

—

y",

(M)]sin(2$)

+

[gy„,(M)

+g',

'(M)]cos(2P)

+

[yy„'(M)

—

y„',

)(M)]]E

.

_E,

.

P), ~o:[=,

'k

(co)E' '

(9)

Pb ~ccsin(4P)sinO .

Therefore, for a normally incident beam, P& ~ is zero.

Thus,

P,

f[y

=P».

We note that even &ft9&0, Pby has a

diff'erent rotational symmetry than

P„

If

M is parallel to z~~

[001],

the SH signal with the (s,s)

combination vanishes. With the (s,

_p)

combination, we find

p,

,

(M) =y)2),

,

(M)E2

+2y'

„', (M)E„E,

+yy,

',

(M)E,

=

—

_y'„2,

_)(M)E

_sin(2)9'),

₍₆₎

with the constant term arising from the magnetized

sur-face. Thus from the rotational anisotropy, the contribu-tion

of

the magnetized (001) surface can be uniquely determined.

For M=O,

the SH intensity,

I(2'),

is

pro-portional to sin

(4P)

and exhibits an eightfold symmetry.

For

M&0,

I(2')

assumes a fourfold symmetry and its minima no longer vanish.

B.

(110)surface

For

the case

of

M~~[001], the (s,s) polarization

com-bination yields

P,

~

=

[(2gyy

(M)

—

y'

'(M)]sing

—

_[y'

_„(M)+2y'

'

_(M)

—

y„'

( M)]sin(3$)₎

E

~,

where

E

isthe electric field amplitude and g„'y,

(M)

is an odd function

of M.

As one would expect physically,

P„

is rotationally isotropic with respect to

z.

The quadru-pole contribution from the bulk is again given by

Eq.

(5). The total SH field as a function

of

P is

E,

(2') =

Ci

+

C2sin(4$),

—

_k2,.(co)E.

_E,

.

+

k .

_E,

_{]g sin(2$)}

—

k (co

)E

~/sin(4$

),

(10)

where _k2,(co) is the z component

of

the wave vector

of

the refracted light in the magnetic material. In the ab-sence

of

surface magnetization,

y',

)_=y„' _',

_&0,

_while

,

=0.

As a result,

P,

~

=0.

The re~ecte

(2) (2) yyz xxz

signal is contributed solely by the bulk.

For

M&0,

it is possible to identify unambiguously the SH signal due to

the magnetized surface by employing geometries with

/=0

or

P=a.

/2, at which Pb

=0,

according to Eqs. (9) and (10), while

_P,

=y'„,

)(M)E

sin(28') for

/=0,

and

P,

=

—

y',

)(M)E

sin(28') for P=m.

/2.

C. (111)surface

and

Pb ~

~ gE sin(3$)

.

Examining Eqs

(11)

and (12), we fnd

P,

y =Xy(2yy)(M )Ey2

while Pb ~

=0

for

/=0.

Both the surface and bulk

con-tributions to the nonlinear polarization are identically zero in the absence

of

surface magnetization for tt)

=0,

but the former isnonvanishing for

M&0.

For

Malong the

[111]

axis, the (s,s) polarization

com-bination gives

Let us consider the (s,s)polarization combination.

For

M~~[2 1

1],

i.

e.

,M in the mirror plane, we have

P,

=[

—

y„',' _(M)sin

P

—

y'

'(M)singcos

P

—

2y' „'(M)sing cos

P+2y'„~

(M)sin P

+y',

' _(M)sin

P

cosP+y'

(M)cos

P]E

~

(7) which is nonvanishing only for

M&0.

The equivalent surface nonlinear polarization due tothe bulk quadrupole

1s

P,

y

=[y'

'

_(M)sin(3$)+y

_(M)cos(3$)]E

~ Pb ~

~k2,

.(co)gE

sin(3$)

. (13) (14)

(5)

symme-try about

[111].

The former, induced by M, can be unambiguously determined by choosing

/=0.

IV. THK MAGNETIZATION-INDUCED y'

'(M)

In the previous section we have suggested several ex-perimental geometries to probe the surface

magnetiza-tion.

It

is, however, important to know whether the magnetization-induced components

of

y'

'(M)

are large enough to give detectable SH signals. In this section we will estimate these components to the linear order in

M.

Consider Ni(001) as an example. We estimate

y' '

_(M)/y'

«(0)

=0.

1 since it should be

of

the same

or-der as the ratio

of

the linear optical susceptibility

com-ponents

y„' (M)/g('„'(0),

which isabout

0.

1.'

It

was first pointed out by Kittel in an

order-of-magnitude estimate that it isthe change

of

the electronic wave functions, rather than the shift

of

energy eigenval-ues due to spin-orbit coupling, which is responsible for

the magnetization-induced

y,

'"(M)

in a ferromagnetic material. In the calculation for

y„'"(M) of

Ni and

Fe,

Ar-gyres ' showed that this theory does give reasonable

pre-dictions. Here, we _apply the same method to estimate

y„"y(M)/y"„',

(0).

In the electric dipole approximation, the nonlinear sus-ceptibility for

SHG

(Ref.22)is

+5

similar terms

f,

(k),

(15)

where k denotes the electron wave vector; v,c, c' are band

indices forthe valance and conduction bands;

f,

(k)

isthe

Fermi distribution function forthe state ~v,k

);

(»,

)

„,

isa

matrix element for the ith component

of

the electronic

displacement vector for the transition c

~v;

and

(v„=

(E,

E,

)/fi.

—

The Hamiltonian for amagnetized system is

(k,

r)

with

m&n,

i.

e.

,

y„(k,

r)=

g

b„P

(k,

r)

.

m (~n)

(20)

From Eqs. (18)and (19),the expression for y',_~& in

Eq.

(15)can be rewritten as

(M=O)+y'

'(M)

(21) where e

+ +

(» ) (»

i vc

j

)cc'

(»

k &c'c ))I'

„.

..

(2(v

—

~v„)(cv

—

(v,

„)

+

5 similar terms

f,

(k)

(22) and

&»;)„(»,&„(»„&,

,

'„'(M)=

_{g g}

_g+b,

*

+35

similar terms

f,

(k,

(r) . (23)

Here, cr is a spin index, and the

+

signs correspond to

spin up and down, respectively.

If

both spin states

of

~v,

k)

are filled, their contributions in

Eq.

(23) are

can-celed exactly. Thus only the partially filled spin states contribute in the summation.

Calculating the tensor elements in

Eq.

(22) and

Eq.

(23) would require detailed information on the wave

func-tions. Instead, we will calculate only the ratio

of

the magnetization-induced element y(

& (M)

to

the

magnetization-independent nonvanishing element _y,',j,

(0).

We assume all nonvanishing y'_,k)(0) are alike and all

(M)

are alike. From Eqs. (22) and (23),we find

H

=H0+H.

.

p

«(M)/y,

',

q(0)

=

u(,

g

6b„ (24} where and 1 p

+

V(r)

2m c (16) A cl11 '

(E

E

) (25) where ub is the number

of

Bohr magnetons per surface

atom.

We take the approximation by Argyres, '

H,

,

=

1

[VV(r)Xp].

S .

2m c

Treating the spin-orbit coupling Hamiltonian as a

pertur-bation, we find that the electron wave functions with spin functions

a(+1)

are '

P&

+,

=

[g„(k,r)+y„(k,

r)

]a(+1)

and the corresponding eigenenergies are

eg ._(-i

=E

(k)+5

(18)

(19)

where )t

„

is the unperturbed eigenfunction with

HQQ„=E„g„and y„(k,

r)

is a linear combination

of

y' _p«(M)/y',

Jk(0)=0

07i . . (26) This is in excellent agreement with our early estimate based on the known ratio

of

X(yi)(M)/X(„l.

)(0).

Knowing that for a metal surface y',

"k(0}

—

10 ' esu,' we should expect y(

&

(M)

—

10

' _esu. _This _is_certainly detectable, considering that the detection limit

of

surface

SHG isy' '

(

₁₀ ' _esu.'

and consider only the m band just above or below the v

band in the summation, over m.

For

Ni, we take A

=

1.

0 X 10 ' erg, ' the average nearest band separa-tion

of

4

eV=6. 4X10

' erg, ' and

ub=0.

68 for a free surface, we get

(6)

1234 RU-PIN PAN, H.D.WEI, AND

Y.

R.

SHEN 39 V. SUMMARY

We have shown that surface magnetism can lower the surface symmetry and make several elements

of

the

sur-face nonlinear susceptibility tensor for surface SHG no longer vanishing. These elements are derived and tabu-lated for (001),

(110),

and

(111)

surfaces

of

fcc centrosym-metric crystals. They are

of

the order

of

10 ' esu for

nickel, as estimated from the microscopic derivation.

Based on this estimate, it is believed that optical SHG

can be used to probe surface magnetization. Suitable input-output polarization combinations are suggested for

the probing

of

the (001), (110),and

(111)

surfaces with various directions

of M.

It

is also predicted that SHG

signals from the magnetized and nonmagnetized surfaces

in certain experimental geometries will exhibit

character-istically different rotational anisotropy. ACKNOWLEDGMENTS

This work was initiated when Ru-Pin Pan was on leave

at University

of

California at Berkeley, supported by a grant from the National Science Council, Republic

of

China. She would also like to thank members

of

the Physics Department

of

the University

of

California at

Berkeley for their hospitality.

Y.R.

S.

was supported by the Director, Office

of

Energy Research, Office

of

Basic

Energy Sciences, Materials Science Division

of

the U.

S.

Department

of

Energy under Contract No.

DE-AC03-76SF-00098.

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