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Size effect on the low field hall coefficient of Cu and Al single crystals
Physica 10 7B (1981) 501-502 North-Holland Publishing Company
ID 3
SIZE EFFECT ON THE LOW FIELD HALL COEFFICIENT OF Cu AND A1 SINGLE CRYSTALS
Kiyoshi Yonemitsu, Isao Sakamoto and Hideyuki Sato
Department of Physics, Faculty of Science, Tokyo Metropolitan University Fukazawa 2-1-1, Setagayaku, Tokyo, Japan
The s i z e e f f e c t s o f t h e H a l l c o e f f i c i e n t RH o f v e r y p u r e s i n g l e c r y s t a l s o f Cu and A1 a r e m e a s u r e d down t o 3~5 x 10-4T a t 4.2K. The s i g n o f Ru o f Cu samples (B ff [110]) c h a n g e s from n e g a t i v e t o p o s i t i v e a t 0.025T. .RH o f A1 sample (B H [001]) becomes more n e g a t i v e t h a n b u l k sample i n t h e low f i e l d r e g i o n . I t i s s u g g e s t e d t h a t such m e a s u r e ment can be used t o o b t a i n i n f o r m a t i o n s on some l i m i t e d r e g i o n o f Fermi s u r f a c e .
The low field Hall coefficient (RH0) depends both on the relaxation time TCk) and on the curvature 0(k) at every point on the Fermi surface (FS), while the high field ~ depends solely on the FS topology. We cannot extract, however, an information on any part of the FS from RH^ of bulk samples even if it is measured for s~fi~le crystals, since it depends on quantities obtained by the surface integration over FS. As the sample thickness to bulk mean free path ratio,d/£ decreases, contributions from these parts of FS which have large v z (the velocity component normal to the sample surface) fade away, and contributions from the FS parts which have small v z survive. Thus in the size effect measurement of single crystals, we can select the contribution of some local portion by suppressing other parts by artificially settled scatterers: the sample surfaces. Up t o t h e p r e s e n t t i m e , however, o n l y l i m i t e d m e a s u r e m e n t s have been r e p o r t e d f o r p o l y crystals [1 - 3].
J#[IlO] f o r Cu; B # [ 0 0 1 ] , J # [ 1 0 0 ] f o r A1. The l e n g t h and w i d t h o f t h e samples a r e 25 and S mm. The t h i c k n e s s o f samples was d e t e r m i n e d by two d i f f e r e n t ways c o n s i s t e n t l y : t h e measurement o f electrical resistance at room temperature and the measurement of the period of the Sondheimer oscillation. The error in thickness is estimated to be less than 5%. The bulk RRR (residual resistance ratio) of samples is about 20000. Thickness of Cu sample is 118pm and that of A1 is 112um. The corresponding d/~ is about 0.2 for both samples. The magnetic field was produced by a super conducting magnet for B<0.1T and by an iron core electric magnet for B>O.IT. The Hall voltage in the low field region was measured by a model 330 SQUID of SHE Co. Ltd. down to IpV. It was measured by a galvanometer-phototube amplifier system in the high and intermediate field region down to InV.
F i g . 1 shows t h e f i e l d In the experimental point o f view, a difficulty lies in the measurement of very small Hall voltage. In order that the electron-surface scattering prevails over the electron-phonon or electron-impurity scatterings, we must prepare very pure single crystals at very low temperature. Under such conditions x becomes very long and the low field condition ~cT~ I, where w c is the cyclotron frequency, is attainable at very low magnetic field where the Hall voltage is too small to be measured accurately. Recently this difficulty was overcome by the SQUID potentiometer which enables us to measure the voltage as small as 10 "12 volts.
I n the present work we describe the size effect measurement of the R H of Cu and A1 single crystals at 4.2K dowii to 3 ~ x 10-~T. High purity single crystals were oriented by the usual Laue back reflection technique with an accuracy of ±0.SP From the oriented crystals rectangular samples were cut by an acid saw and thinned to desired thickness by an acid polishing and chemical polishing procedures. The sample orientations are: B # [110], 03784363/81/0000-.0000/$02.50
© Not'th-HollandPubRshingCompany
B(T) +
10
d e p e n d e n c e o f 1~ o f t h e
10
10"
2 --~
J
_t_..e_oe __e._e ~ e -.~-e'e'-e" e ' q N ~ ------\6"
[11o3
-4
IC~
- 6
I
-1 < Vz> ...... t_~ ~ k z 1
~ I litll[
I
i_L./!]ll[
1
t [ I I Illl
I
I
1 [ I I11
F i g . 1 The H a l l c o e f f i c i e n t o f a Cu s i n g l e c r y s t a l ( B ff [ 1 1 0 ] ) . The i n s e r t s show schematic p r o j e c t i o n o f the Fermi surface and k z dependence o f v z.
501
502
Cu sample. When B>0.2T, RH is nearly equal to the high field limiting value with a Sondheimer oscillation. As the field strength decreases, RH varies to the positive direction and at 0.025T even its,sign changes. This type of sign reversal of Cu single crystal was first observed in the present work. This behavior is easily understood. As B decreases, most part of the FS die away. The only part that survives is the region that bears the dog's bone orbit on it (the shaded region in Fig.1 insert), since in the region the electron velocity is nearly parallel to the sample surfaces• We may regard the variation below 0.02~0.01T as the "high field-low field transition"of the region. At lower fields, R~ converges to a definite positive value whicW'is about +0.5 x 10-11m3/C.
F i g . 2 shows t h e f i e l d d e p e n d e n c e o f t h e A1 s i n g l e c r y s t a l s . For t h e sake o f c o m p a r i s o n , t h e d a t a o f a t h i c k e r sample (d=91Sum) which i s c u t from t h e same s i n g l e c r y s t a l b l o c k and h a s t h e same o r i e n t a t i o n as t h e t h i n n e r one a r e a l s o p l o t t e d . So as t o p l o t t h e d a t a i n a same f i g u r e t h e h o r i z o n t a l a x i s f o r t h e t h i c k e r sample i s n o r m a l i z e d by m u l t i p l y i n g a f a c t o r q / p ~ where and P2 a r e r e s i s t i v i t i e s o f t h i n ~nd t h i c k ~ samples r e s p e c t i v e l y . In t h e i n t e r m e d i a t e
----I-- I
I I'TTIT"""
!
I
TTTTFTTI---q--I I IIITT~
I
6
RHO 0211(02B)
aE 2dS/lYrEi > all = (eZ/4~3)
4 2
112,m *
--2 ,-,-•-,-•-*-'"~....I_ i i lllliI]O'3i
f;:~'"/
/**~
#$ dd ~SP* 915um
//
B(T)
i ,j~u~lO- 2l
(2)
= (e3Bl4~3h)x
aE
aE r a2E
aE
a2E
.)dS/17JEI>
T(k.r)@-~y(~ -~x2- -~x akx@ky
63) Here E i s t h e e l e c t r o n i c e n e r g y and k . ' s a r e components o f wave v e c t o r . In t h e s e e ~ u a t i o n s , T(k) i s g i v e n by T(k,r) -I : II~ b + 1 /Irlvz(k)[
(4)
where r is the depth of any point in the sample from one of the sample surfaces, and z. is the • . , D bulk relaxation tlme whlch is taken to be constant. The brackets in Eqs. (2) and (3) mean the average taken over r. The calculated and I
I
I I IIII
I
I
RH - l (lO-I ]m3/C?
_-3
+ RH (]0-Iim3/C)
(I)
:
where
I
I IIIll
(a)
(b) '~
I I frrT)
-10 8
basis of 4-OPW approximation•
-1 I JJIlJll ] 0 , i llILLt
F i g . 2 The H a l l C o e f f i c i e n t o f two AI s i n g l e c r y s t a l s ( B / / [ 0 0 1 ] ) . The h o r i z o n t a l a x i s for thicker sample is normalized (see text). f i e l d , RH o f t h i n n e r sample i s more p o s i t i v e . As B d e c r e a s e s b o t h c u r v e s c r o s s zero n e a r l y a t O.O1T. I n t h e l o w e r f i e l d r e g i o n , RH o f t h i n n e r sample becomes more n e g a t i v e . The size effect of A1 is not so easily under= s t o o d as in the case of Cu, since FS of A1 is composed of two sheets: the 2nd zone hole sheet and t h e 3 r d zone electron sheet which have very complicated shape. The bulk RH0 is a result of critical balance between the contributions from
these sheets. In order to gain an insight into this problem, we have computed the size dependent RH0 on the
/~ ÷
0.5 1.0 I ~ i~lllll
5.0 I I l lllllJ
Doo]
+
Fig.3 (a) The computed (a s o l i d c u r v e ) and t h e measured low f i e l d H a l l c o e f f i c i e n t o f two A1 s a m p l e s ( - @ - ) . (b) S t e r e o g r a p h i c p r o j e c t i o n o f A1 2nd zone Fermi s h e e t . D o t t e d a r e a makes h o l e l i k e c o n t r i b u t i o n . The r e g i o n t h a t s u r v i v e s i n t h i n sample i s shown by a shaded area. measured Run are plotted in Fig.3(a). The agreement b e t w e ~ them is good. It was also known from the calculation that in this orientation, most parts of the FS are suppressed to the same extent by the surface scattering except a shaded area in Fig.3(b). This shows that the dominant contribution to the measured RH0 comes from this part. Finally we suggest that the size effect of the low field RH in single crystals can be a tool to study some specific part of Fermi surface. REFERENCES
:
[I] S u r i R i t u , T h a k o o r , A . P . and C h o p r a , K . L . , J . Appl. Phys. 46 (1975) 2574-82. [2] P~rsvoll,K. and Holwech, I., Phil. Mag. 10 (1964) 921-30. [3] Cooper,J.N., Cotti,P. and Rasmussen, F.B., Phys.Letters 19 (1965) 560-62.
ID 3
SIZE EFFECT ON THE LOW FIELD HALL COEFFICIENT OF Cu AND A1 SINGLE CRYSTALS
Kiyoshi Yonemitsu, Isao Sakamoto and Hideyuki Sato
Department of Physics, Faculty of Science, Tokyo Metropolitan University Fukazawa 2-1-1, Setagayaku, Tokyo, Japan
The s i z e e f f e c t s o f t h e H a l l c o e f f i c i e n t RH o f v e r y p u r e s i n g l e c r y s t a l s o f Cu and A1 a r e m e a s u r e d down t o 3~5 x 10-4T a t 4.2K. The s i g n o f Ru o f Cu samples (B ff [110]) c h a n g e s from n e g a t i v e t o p o s i t i v e a t 0.025T. .RH o f A1 sample (B H [001]) becomes more n e g a t i v e t h a n b u l k sample i n t h e low f i e l d r e g i o n . I t i s s u g g e s t e d t h a t such m e a s u r e ment can be used t o o b t a i n i n f o r m a t i o n s on some l i m i t e d r e g i o n o f Fermi s u r f a c e .
The low field Hall coefficient (RH0) depends both on the relaxation time TCk) and on the curvature 0(k) at every point on the Fermi surface (FS), while the high field ~ depends solely on the FS topology. We cannot extract, however, an information on any part of the FS from RH^ of bulk samples even if it is measured for s~fi~le crystals, since it depends on quantities obtained by the surface integration over FS. As the sample thickness to bulk mean free path ratio,d/£ decreases, contributions from these parts of FS which have large v z (the velocity component normal to the sample surface) fade away, and contributions from the FS parts which have small v z survive. Thus in the size effect measurement of single crystals, we can select the contribution of some local portion by suppressing other parts by artificially settled scatterers: the sample surfaces. Up t o t h e p r e s e n t t i m e , however, o n l y l i m i t e d m e a s u r e m e n t s have been r e p o r t e d f o r p o l y crystals [1 - 3].
J#[IlO] f o r Cu; B # [ 0 0 1 ] , J # [ 1 0 0 ] f o r A1. The l e n g t h and w i d t h o f t h e samples a r e 25 and S mm. The t h i c k n e s s o f samples was d e t e r m i n e d by two d i f f e r e n t ways c o n s i s t e n t l y : t h e measurement o f electrical resistance at room temperature and the measurement of the period of the Sondheimer oscillation. The error in thickness is estimated to be less than 5%. The bulk RRR (residual resistance ratio) of samples is about 20000. Thickness of Cu sample is 118pm and that of A1 is 112um. The corresponding d/~ is about 0.2 for both samples. The magnetic field was produced by a super conducting magnet for B<0.1T and by an iron core electric magnet for B>O.IT. The Hall voltage in the low field region was measured by a model 330 SQUID of SHE Co. Ltd. down to IpV. It was measured by a galvanometer-phototube amplifier system in the high and intermediate field region down to InV.
F i g . 1 shows t h e f i e l d In the experimental point o f view, a difficulty lies in the measurement of very small Hall voltage. In order that the electron-surface scattering prevails over the electron-phonon or electron-impurity scatterings, we must prepare very pure single crystals at very low temperature. Under such conditions x becomes very long and the low field condition ~cT~ I, where w c is the cyclotron frequency, is attainable at very low magnetic field where the Hall voltage is too small to be measured accurately. Recently this difficulty was overcome by the SQUID potentiometer which enables us to measure the voltage as small as 10 "12 volts.
I n the present work we describe the size effect measurement of the R H of Cu and A1 single crystals at 4.2K dowii to 3 ~ x 10-~T. High purity single crystals were oriented by the usual Laue back reflection technique with an accuracy of ±0.SP From the oriented crystals rectangular samples were cut by an acid saw and thinned to desired thickness by an acid polishing and chemical polishing procedures. The sample orientations are: B # [110], 03784363/81/0000-.0000/$02.50
© Not'th-HollandPubRshingCompany
B(T) +
10
d e p e n d e n c e o f 1~ o f t h e
10
10"
2 --~
J
_t_..e_oe __e._e ~ e -.~-e'e'-e" e ' q N ~ ------\6"
[11o3
-4
IC~
- 6
I
-1 < Vz> ...... t_~ ~ k z 1
~ I litll[
I
i_L./!]ll[
1
t [ I I Illl
I
I
1 [ I I11
F i g . 1 The H a l l c o e f f i c i e n t o f a Cu s i n g l e c r y s t a l ( B ff [ 1 1 0 ] ) . The i n s e r t s show schematic p r o j e c t i o n o f the Fermi surface and k z dependence o f v z.
501
502
Cu sample. When B>0.2T, RH is nearly equal to the high field limiting value with a Sondheimer oscillation. As the field strength decreases, RH varies to the positive direction and at 0.025T even its,sign changes. This type of sign reversal of Cu single crystal was first observed in the present work. This behavior is easily understood. As B decreases, most part of the FS die away. The only part that survives is the region that bears the dog's bone orbit on it (the shaded region in Fig.1 insert), since in the region the electron velocity is nearly parallel to the sample surfaces• We may regard the variation below 0.02~0.01T as the "high field-low field transition"of the region. At lower fields, R~ converges to a definite positive value whicW'is about +0.5 x 10-11m3/C.
F i g . 2 shows t h e f i e l d d e p e n d e n c e o f t h e A1 s i n g l e c r y s t a l s . For t h e sake o f c o m p a r i s o n , t h e d a t a o f a t h i c k e r sample (d=91Sum) which i s c u t from t h e same s i n g l e c r y s t a l b l o c k and h a s t h e same o r i e n t a t i o n as t h e t h i n n e r one a r e a l s o p l o t t e d . So as t o p l o t t h e d a t a i n a same f i g u r e t h e h o r i z o n t a l a x i s f o r t h e t h i c k e r sample i s n o r m a l i z e d by m u l t i p l y i n g a f a c t o r q / p ~ where and P2 a r e r e s i s t i v i t i e s o f t h i n ~nd t h i c k ~ samples r e s p e c t i v e l y . In t h e i n t e r m e d i a t e
----I-- I
I I'TTIT"""
!
I
TTTTFTTI---q--I I IIITT~
I
6
RHO 0211(02B)
aE 2dS/lYrEi > all = (eZ/4~3)
4 2
112,m *
--2 ,-,-•-,-•-*-'"~....I_ i i lllliI]O'3i
f;:~'"/
/**~
#$ dd ~SP* 915um
//
B(T)
i ,j~u~lO- 2l
(2)
= (e3Bl4~3h)x
aE
aE r a2E
aE
a2E
.)dS/17JEI>
T(k.r)@-~y(~ -~x2- -~x akx@ky
63) Here E i s t h e e l e c t r o n i c e n e r g y and k . ' s a r e components o f wave v e c t o r . In t h e s e e ~ u a t i o n s , T(k) i s g i v e n by T(k,r) -I : II~ b + 1 /Irlvz(k)[
(4)
where r is the depth of any point in the sample from one of the sample surfaces, and z. is the • . , D bulk relaxation tlme whlch is taken to be constant. The brackets in Eqs. (2) and (3) mean the average taken over r. The calculated and I
I
I I IIII
I
I
RH - l (lO-I ]m3/C?
_-3
+ RH (]0-Iim3/C)
(I)
:
where
I
I IIIll
(a)
(b) '~
I I frrT)
-10 8
basis of 4-OPW approximation•
-1 I JJIlJll ] 0 , i llILLt
F i g . 2 The H a l l C o e f f i c i e n t o f two AI s i n g l e c r y s t a l s ( B / / [ 0 0 1 ] ) . The h o r i z o n t a l a x i s for thicker sample is normalized (see text). f i e l d , RH o f t h i n n e r sample i s more p o s i t i v e . As B d e c r e a s e s b o t h c u r v e s c r o s s zero n e a r l y a t O.O1T. I n t h e l o w e r f i e l d r e g i o n , RH o f t h i n n e r sample becomes more n e g a t i v e . The size effect of A1 is not so easily under= s t o o d as in the case of Cu, since FS of A1 is composed of two sheets: the 2nd zone hole sheet and t h e 3 r d zone electron sheet which have very complicated shape. The bulk RH0 is a result of critical balance between the contributions from
these sheets. In order to gain an insight into this problem, we have computed the size dependent RH0 on the
/~ ÷
0.5 1.0 I ~ i~lllll
5.0 I I l lllllJ
Doo]
+
Fig.3 (a) The computed (a s o l i d c u r v e ) and t h e measured low f i e l d H a l l c o e f f i c i e n t o f two A1 s a m p l e s ( - @ - ) . (b) S t e r e o g r a p h i c p r o j e c t i o n o f A1 2nd zone Fermi s h e e t . D o t t e d a r e a makes h o l e l i k e c o n t r i b u t i o n . The r e g i o n t h a t s u r v i v e s i n t h i n sample i s shown by a shaded area. measured Run are plotted in Fig.3(a). The agreement b e t w e ~ them is good. It was also known from the calculation that in this orientation, most parts of the FS are suppressed to the same extent by the surface scattering except a shaded area in Fig.3(b). This shows that the dominant contribution to the measured RH0 comes from this part. Finally we suggest that the size effect of the low field RH in single crystals can be a tool to study some specific part of Fermi surface. REFERENCES
:
[I] S u r i R i t u , T h a k o o r , A . P . and C h o p r a , K . L . , J . Appl. Phys. 46 (1975) 2574-82. [2] P~rsvoll,K. and Holwech, I., Phil. Mag. 10 (1964) 921-30. [3] Cooper,J.N., Cotti,P. and Rasmussen, F.B., Phys.Letters 19 (1965) 560-62.