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Computer simulation of the imaging of magnetic domains in the scanning electron microscope
C O M P U T E R S I M U L A T I O N O F T H E I M A G I N G O F M A G N E T I C D O M A I N S IN T H E S C A N N I N G ELECTRON MICROSCOPE J. P. J A K U B O V I C S Dept. of Metallurgy and Science of Materials, Parks Road, Oxford OX1 3PH, UK
and O.C. W E L L S I B M Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA
The imaging of magnetic domains in the scanning electron microscope has been investigated by computer simulation. The results show that information about the magnetization is carried only by low energy-loss electrons. Unwanted background could therefore be removed by filtering out low energy electrons. Energy filtering also results in increased resolution.
1. Introduction Several methods are currently in use for imaging magnetic domains in the scanning electron microscope (SEM) [1]. One of these methods may be used to image domains at high resolution [2-4]. The image contrast and resolution has been studied previously by computer simulation [5]. The results of these calculations can be correlated with images formed by using the absorbed electron count as the signal. However, the image formed with the backscattered electron count as the signal is not necessarily complementary to the absorbed electron image, since backscattered electron detectors capture only electrons travelling within a certain solid angle of directions. The image may also be modified by filtering out backscattered electrons with energies in a certain range. These factors may be exploited in producing images of improved quality by a suitable choice of detector and its position, and by suitable energy filtering. In this paper, we describe calculations aimed at investigating these possibilities.
2. Method of calculation The computer program was based on that used previously [5]. In the present case, a specimen of thickness t was assumed. It was assumed to be divided into two domains magnetized parallel to the surface, divided by a 180 ° wall perpendicular to the surface, and to be attached to a substrate of identical material but with zero magnetization. (This may be a reasonable model for overlay films in bubble domain devices.) Electrons were assumed
to be incident on the surface in a specified direction, at a specified distance from the domain wall. The number of electrons backscattered with various energies in various directions was counted.
3. Results Fig. 1 shows the variation of "0E, the ratio of the number of electrons backscattered per unit energy range to n, the number of incident electrons, as a function of energy, E, for two opposite directions of magnetization, as well as the difference between them. In the caption, E o is the incident beam energy, O is the angle of incidence, R is the electron range (see ref. [5] for values), s is the number of scattering events along a path of length R, and r is the ratio of the saturation magnetic induction used in the calculation to BEe, the saturation induction of iron (BFe = 2.16 T). It is seen that the difference is a maximum at an energy of E m ~ 27.4 keV ( E m / E o ~ 0.91), and is essentially zero for E z 20 keY ( E z / E o ~ 0 . 6 7 ) . These values are unchanged down to t = 0.5 /tm ( t / R ~ 0.15). However, R increases with increasing E o [5], and we should therefore expect an increase of E m / E o and E J E o with increasing Eo, for constant t. For t -0.5 /tm, values found were E m / E o ~ 0 . 9 2 5 and E z / E o ~ 0.73 for g o = 60 keY ( t / R ~ 0.0425), and E m / E o ~ 0 . 9 5 3 and E J E o ~ 0.8 for E o = 100 kV ( t / R ~ 0.0172). For t = 0.2 /~m, we find E m / E o ~ 0.948 and E z / E o ~ 0.82 at E o = 60 keV ( t / R ~ 0.0174). This suggests that it should be advantageous to filter out the lower energy electrons. If Ef is the highest energy to be filtered out,
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1523-1525 ©North Holland
152,3
J. P. Jakubovics, O. C. Wells~Imaging of magnetic domains in the SEM
1524
E r / E o should increase with decreasing t at a given
r[(o)
E o, or with increasing Eo at a given t. The above calculations assumed a detector subtending a solid angle of 2~r above the specimen. Calculations were also carried out for a detector subtending a smaller solid angle. In order to specify the size and position of the detector, we use the polar coordinates ~, ~', where
0 436
xL(-) o
,3 4 3 5
0
o 0 434
40.194
Oo oo
0 433.
'0 ~93
o o
0-432 o 0 431[
= cos-~(i's),
s) x i ) . j ] / l i x
= cos-l{[((/x
sl};
•o
o
i0192
o
is
•
-10
0
-5
5 × (p.m)
i, j and s are unit vectors normal to the plane of incidence, along the intersection of the plane of incidence with the specimen surface, and along a line from the origin to a given point on the detector, respectively. If the centre of the detector is at 90°, ~ ~'t, then ~'t is called the take-off angle. In the calculations, a detector bounded by ~ = (90 - 20) ° , ~" fit -+- 50 was assumed, and plots of the type shown in fig. 1 were obtained for various values of ~t. As expected, both E m / E o and E z / E o decrease with increasing ~'t, since the electrons must on average travel a longer distance inside the specimen before being scattered by large angles. The number of electrons collected is a maximum for ~'t ~ 45°; for this take-off angle and t >> R, E m / E o 0.923 and E z / E o ~ 0.700. =
195
oo
--
=
Fig. 2. Variation of r/, the ratio of the n u m b e r of electrons collected to the n u m b e r of incident electrons, with x, the distance of the point of i n c i d e n c e from the d o m a i n wall, calculated for Eo = lOOkeV, O = 4 5 ° , t > > R , n = 5 x 104, s = 100, r = 1, a n d a solid angle of detection of 2~r. Left-hand ordinate axis a n d O refers to Ef = 0, right-hand ordinate axis and • refers to Ef = 80 keV.
Since these results all suggest the desirability of filtering out low-energy backscattered electrons, the effect of energy filtering on image profiles [5] was investigated. In each case, it was found that the image profiles became narrower with increasing El. Fig. 2 shows a comparison between an unfiltered image and an image corresponding to Ef 0.8Eo, close to E~ for this case. For Ef = 0, the average value of T/, the ratio of the number of
x10-~, 5"T
>
aj ~ , L, :!3~a
4+ MrP
v
3!
~
Z~r, q?
..
n*
• 'da ' :1 •
I
•
i
c
m"
.
• t~,
T
IT ,
M
÷
0 ~-
O
I0
20
30 E(keV)
Fig.
1. Variation of I'/E with E, for E o = 30 keV, 0 = 45 °, t :~ R, n - 2 x 105, s = 100, r = 50. T h e values c o r r e s p o n d i n g to the two d o m a i n s are d e n o t e d b y x a n d I-1, respective_ly, a n d + s h o w s the difference.
j . p. Jakubovics, O. C. Wells/Imaging of magnetic domains in the S E M
IS25
electrons collected to the n u m b e r of i n c i d e n t electrons, is 0.43, the c o n t r a s t b e t w e e n the two d o m a i n s , C, is 1.1% a n d the i m a g e profile width, X [5], is 3.64/~m. F o r Ef ~ 0 . 8 E o, the a v e r a g e value of ~/ is 0.19, C is 2.0% a n d X is 1.82/tm.
values in each case, a n d m o r e d e t a i l e d c a l c u l a t i o n s are b e i n g p l a n n e d .
4. Conclusions
[1] D. J. Fathers and J. P. Jakubovics, Physica 86-88B (1977) 1343. [2] D. J. Fathers, J. P. Jakubovics, D. C. Joy, D. E. Newbury and H. Yakowitz, Phys. Stat. Sol. (a) 20 (1973) 535. [3] D. J. Fathers, J. P. Jakubovics, D. C. Joy, D. E. Newbury and H. Yakowitz, Phys. Stat. Sol. (a) 22 (1974) 609. [4] D. J. Fathers and J. P. Jakubovics, Phys. Stat. Sol. (a) 36 (1976) K13. [5] J. P. Jakubovics and D. J. Fathers, Phys. Stat. Sol. (a) 46 (1978) 291.
T h e results suggest that e n e r g y filtering s h o u l d i m p r o v e the images of d o m a i n s . I n o r d e r to o b t a i n the best images, b o t h E o a n d Ef s h o u l d be optimized. O p t i m u m values will also d e p e n d o n t, a n d on the d o m a i n size. T h e m e t h o d u s e d in the present calculations c o u l d be u s e d to p r e d i c t o p t i m u m
References
and O.C. W E L L S I B M Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA
The imaging of magnetic domains in the scanning electron microscope has been investigated by computer simulation. The results show that information about the magnetization is carried only by low energy-loss electrons. Unwanted background could therefore be removed by filtering out low energy electrons. Energy filtering also results in increased resolution.
1. Introduction Several methods are currently in use for imaging magnetic domains in the scanning electron microscope (SEM) [1]. One of these methods may be used to image domains at high resolution [2-4]. The image contrast and resolution has been studied previously by computer simulation [5]. The results of these calculations can be correlated with images formed by using the absorbed electron count as the signal. However, the image formed with the backscattered electron count as the signal is not necessarily complementary to the absorbed electron image, since backscattered electron detectors capture only electrons travelling within a certain solid angle of directions. The image may also be modified by filtering out backscattered electrons with energies in a certain range. These factors may be exploited in producing images of improved quality by a suitable choice of detector and its position, and by suitable energy filtering. In this paper, we describe calculations aimed at investigating these possibilities.
2. Method of calculation The computer program was based on that used previously [5]. In the present case, a specimen of thickness t was assumed. It was assumed to be divided into two domains magnetized parallel to the surface, divided by a 180 ° wall perpendicular to the surface, and to be attached to a substrate of identical material but with zero magnetization. (This may be a reasonable model for overlay films in bubble domain devices.) Electrons were assumed
to be incident on the surface in a specified direction, at a specified distance from the domain wall. The number of electrons backscattered with various energies in various directions was counted.
3. Results Fig. 1 shows the variation of "0E, the ratio of the number of electrons backscattered per unit energy range to n, the number of incident electrons, as a function of energy, E, for two opposite directions of magnetization, as well as the difference between them. In the caption, E o is the incident beam energy, O is the angle of incidence, R is the electron range (see ref. [5] for values), s is the number of scattering events along a path of length R, and r is the ratio of the saturation magnetic induction used in the calculation to BEe, the saturation induction of iron (BFe = 2.16 T). It is seen that the difference is a maximum at an energy of E m ~ 27.4 keV ( E m / E o ~ 0.91), and is essentially zero for E z 20 keY ( E z / E o ~ 0 . 6 7 ) . These values are unchanged down to t = 0.5 /tm ( t / R ~ 0.15). However, R increases with increasing E o [5], and we should therefore expect an increase of E m / E o and E J E o with increasing Eo, for constant t. For t -0.5 /tm, values found were E m / E o ~ 0 . 9 2 5 and E z / E o ~ 0.73 for g o = 60 keY ( t / R ~ 0.0425), and E m / E o ~ 0 . 9 5 3 and E J E o ~ 0.8 for E o = 100 kV ( t / R ~ 0.0172). For t = 0.2 /~m, we find E m / E o ~ 0.948 and E z / E o ~ 0.82 at E o = 60 keV ( t / R ~ 0.0174). This suggests that it should be advantageous to filter out the lower energy electrons. If Ef is the highest energy to be filtered out,
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1523-1525 ©North Holland
152,3
J. P. Jakubovics, O. C. Wells~Imaging of magnetic domains in the SEM
1524
E r / E o should increase with decreasing t at a given
r[(o)
E o, or with increasing Eo at a given t. The above calculations assumed a detector subtending a solid angle of 2~r above the specimen. Calculations were also carried out for a detector subtending a smaller solid angle. In order to specify the size and position of the detector, we use the polar coordinates ~, ~', where
0 436
xL(-) o
,3 4 3 5
0
o 0 434
40.194
Oo oo
0 433.
'0 ~93
o o
0-432 o 0 431[
= cos-~(i's),
s) x i ) . j ] / l i x
= cos-l{[((/x
sl};
•o
o
i0192
o
is
•
-10
0
-5
5 × (p.m)
i, j and s are unit vectors normal to the plane of incidence, along the intersection of the plane of incidence with the specimen surface, and along a line from the origin to a given point on the detector, respectively. If the centre of the detector is at 90°, ~ ~'t, then ~'t is called the take-off angle. In the calculations, a detector bounded by ~ = (90 - 20) ° , ~" fit -+- 50 was assumed, and plots of the type shown in fig. 1 were obtained for various values of ~t. As expected, both E m / E o and E z / E o decrease with increasing ~'t, since the electrons must on average travel a longer distance inside the specimen before being scattered by large angles. The number of electrons collected is a maximum for ~'t ~ 45°; for this take-off angle and t >> R, E m / E o 0.923 and E z / E o ~ 0.700. =
195
oo
--
=
Fig. 2. Variation of r/, the ratio of the n u m b e r of electrons collected to the n u m b e r of incident electrons, with x, the distance of the point of i n c i d e n c e from the d o m a i n wall, calculated for Eo = lOOkeV, O = 4 5 ° , t > > R , n = 5 x 104, s = 100, r = 1, a n d a solid angle of detection of 2~r. Left-hand ordinate axis a n d O refers to Ef = 0, right-hand ordinate axis and • refers to Ef = 80 keV.
Since these results all suggest the desirability of filtering out low-energy backscattered electrons, the effect of energy filtering on image profiles [5] was investigated. In each case, it was found that the image profiles became narrower with increasing El. Fig. 2 shows a comparison between an unfiltered image and an image corresponding to Ef 0.8Eo, close to E~ for this case. For Ef = 0, the average value of T/, the ratio of the number of
x10-~, 5"T
>
aj ~ , L, :!3~a
4+ MrP
v
3!
~
Z~r, q?
..
n*
• 'da ' :1 •
I
•
i
c
m"
.
• t~,
T
IT ,
M
÷
0 ~-
O
I0
20
30 E(keV)
Fig.
1. Variation of I'/E with E, for E o = 30 keV, 0 = 45 °, t :~ R, n - 2 x 105, s = 100, r = 50. T h e values c o r r e s p o n d i n g to the two d o m a i n s are d e n o t e d b y x a n d I-1, respective_ly, a n d + s h o w s the difference.
j . p. Jakubovics, O. C. Wells/Imaging of magnetic domains in the S E M
IS25
electrons collected to the n u m b e r of i n c i d e n t electrons, is 0.43, the c o n t r a s t b e t w e e n the two d o m a i n s , C, is 1.1% a n d the i m a g e profile width, X [5], is 3.64/~m. F o r Ef ~ 0 . 8 E o, the a v e r a g e value of ~/ is 0.19, C is 2.0% a n d X is 1.82/tm.
values in each case, a n d m o r e d e t a i l e d c a l c u l a t i o n s are b e i n g p l a n n e d .
4. Conclusions
[1] D. J. Fathers and J. P. Jakubovics, Physica 86-88B (1977) 1343. [2] D. J. Fathers, J. P. Jakubovics, D. C. Joy, D. E. Newbury and H. Yakowitz, Phys. Stat. Sol. (a) 20 (1973) 535. [3] D. J. Fathers, J. P. Jakubovics, D. C. Joy, D. E. Newbury and H. Yakowitz, Phys. Stat. Sol. (a) 22 (1974) 609. [4] D. J. Fathers and J. P. Jakubovics, Phys. Stat. Sol. (a) 36 (1976) K13. [5] J. P. Jakubovics and D. J. Fathers, Phys. Stat. Sol. (a) 46 (1978) 291.
T h e results suggest that e n e r g y filtering s h o u l d i m p r o v e the images of d o m a i n s . I n o r d e r to o b t a i n the best images, b o t h E o a n d Ef s h o u l d be optimized. O p t i m u m values will also d e p e n d o n t, a n d on the d o m a i n size. T h e m e t h o d u s e d in the present calculations c o u l d be u s e d to p r e d i c t o p t i m u m
References