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Longitudinal magnetoresistance of thin metal films
7hm S,Ikl/-Thai,. 57' (1979) 1,1 1,4 ~" Elsevier Sequoia S.A., Lausanne
Printed in the Netherlands
L1
Letter
Longitudinal magnetoresistance of thin metal films ( . R. II.II,I,IER, A. J. IOSSI~R A N I ) ( ' . PI('HARD
Univer.~it~; de Nancy 1. l,ahoratoire d'E/ectronique, ('.0. 140. 54037 N a m y COdex t k)'ume / ( Recei~ ed October 16. 1978 : accepted Oct ober 31, 1978)
1. Introduction In recent years it has been realized that the Hall coefficient R m. of thin metallic films subjected to both magnetic and electric fields will be strongly moditied ~-~'when the contribution to R m of structural defects such as grain boundaries "~ ~' cannot be regarded as negligible: thus G o l m a y o and Sacedon 't and Duggal and Nagpal 6 have suggested that the marked departure from the classical Fuchs-Sondheimer (FS) theory ~ of the thickness dependence of the Hall coefficient in metal films is due to the polycrystalline structure of these films. A suitable expression for the resistivity Pvp of a thin polycrystalline film in the absence of a magnetic field has recently been proposed by Mayadas and Shatzkcs~: they have assumed that the grain boundaries can be represented by two types of randomly spaced planes perpendicular and parallel rcspectively to the electric field and that the electronic scattering on parallel boundaries is only specular so that the problem reduces to a one-dimensional problem. In the case of a transverse magnetic field H: we have to consider applying the Sondheimer model" to two electric field components /='~ and E~.: hence in the Mayadas Shatzkes (MS) model we have to take into account two types of grain boundaries perpendicular to E.~ and E~. and the problem becomes very difficult to solve. Howevcr, in the case of a polycrystalline film subjected (Fig. 1) to both a longitudinal magnetic field (H,., O, O) and an electric field (E~. O, O) we may regard the magnctic field H as only modifying the electron trajectories :.'~.~°: the electrons move in helical paths and will also be scattered by the planes representing the grain boundaries which are parallel to the electric field E x (Fig. 1): as in the MS model we may assume that the electronic scattering from these planes is completely specular and can be neglected.
2. Theoretical It has been shown previously that the simultaneous grain boundary and background scattering effects in the MS model, with the geometry of Fig. 2, can be described by an effective relaxation time ref r which is given by;" J t r~.ff- = T0
+ ~fk~,l
(la)
where ~o refers to the background relaxation time, kF to the magnitude of the Fermi
L2
I.EITIiRS
//_._/// T [
,,
....
f f - qzJl ff
/
r'
IZ . . . . . . .
B
.... °°°+ ..... ,
~
Ex ~
;
'' \ ' ............. 'o
Fig. I. t h e geometry of the thin lihn and the applied lields, r = r o is taken to be the point B, BC is the projection of the helical path of an electron thal has been scattered at point B and c/; is the angle between the projection of the velocity vector v in the (y, z) plane and a fixed direction in this plane {taken parallel to the Oy axis in this easel. Fig 2. The geometry of the grain boundary model.
wavcvector a n d kx to the x c o m p o n e n t of the wavevector. F o r an electron travelling at an angle 0 to the x axis the k., c o m p o n e n t is k., = kt: cos 0
( 1b)
The physical p a r a m e t e r :t is related to the grain b o u n d a r y reflection coefficient r, the average grain size % and the b a c k g r o u n d mean free path Io b y ; :t =
It)
r
a~ l
--
(2)
r
Furthermore it has been shown 11.12 that the total solution of the external surfaces and grain boundary effects can be simply obtained on the basis of the FS theory. The method applies also to the +'kinetic model" developed by Chambers 1'' which describes the external size effect in a thin film. In the absence of a magnetic field and in the case of diffuse scattering on external surfaces the deviationJ~(k) of the d i s t r i b u t i o n functionJ~)(k) of the electrons is .ll(k.r)=
-
m
-
l-exp
-
13)
where t is the speed Ivl of the electrons and r - r o is a vector parallel to v (i.e. to k) which is related to electrons passing t h r o u g h a point r in the m e t a l : ro c o r r e s p o n d s to a point on the external surface of the tilm. On t a k i n g the total drift current and integrating over the cross-sectional area S the tilm c o n d u c t i v i t y o v is given as usual, in the absence of grain boundaries, b v ~°
O'l- ----
--, - , . ' ~ oh3S
f
o
+..i (~I o d r ~'V
£f£ dS
:~ dO
o
~' d 0 s i n 0 c o s 2 0 x
Hence the thin polycrystalline tilm c o n d u c t i v i t y %p m a y bc rewritten from eqn. (4) by substituting ~ft for to:
LETTERS
L3
2earn2
r 3 ~1o dr
o,.~ = 7.~S
dS
,r
de
d0 sin
,,
0 cos20x {5)
Equation (4) is still applicable in the case of thin fihns subjectcd to a longitudinal magnetic field provided that J r - r o t is taken to be the distance measured along the spiral trajectory of the electrons ~". Chambers' o has cvaluated integral {4) by noting that when an electron traverses from r to ro its projection on the I)'. z} plane moves an angle ¢ around a circle of radius R {Fig. 1) whcrc R =-
tHl"
elt.~ s i n 0 =
R osin0
{6)
The geometrical evaluation of [ r - r 0 ] by Chambers 1° is also valid for polycrystalline films and the distance J r - r o l may be written, for the geometry of Fig. I, [r-r,,[ =
Rotl,(y,z,~.Ol
{7)
The introduction ofeqns. (l), (6) and {7) in eqn. (5) gives a general expression for the polycrystalline film conductivity in the presence of a longitudinal magnetic field H~ :
avP
--
2e2m2"r°
x
h'~S
,
z'' ,'~ll, z- dr dS cr d.s
[
{
-¢R°_l+
l-exp
,
dq~
dO sin0c°s20 ] +~[cos0[ 1 x {8)
~,,, ~
Icos0l
Numerical evaluations of ¢ and ofeqn. {8} are somewhat complicated even if Kao's" formulation o r e in terms ofz, 0 and 4~ is used. It should be noted that eqn. [8) satislies the physical requirement that in the limit ~ = 0{no grain boundary effect} it reduces to the Chambers TM formulation. In the case of polycrystalline films with a constant grain size ag (thickness independent) the MS expression for the conductivity is deducedU from the FS expression by substituting the grain boundary' conductivity o-~ for the bulk conductivity a 0` where as is the conductivity of an intinitely thick polycrystalline film ~. Hence the grain boundary conductivity' c is given by eqn. [8) when Rc~ and S have intinite values:
~,=
2 e"23m- 2 ~r ° f * rh.3d~, l ° f : d"e; )
( 'l"
)
x c°s20 0:rlO sin /
)
dO.+-,c:sO~_,
(9)
Equation (9) is in agreement with the expression previously derived by Mayadas and Shatzkes 8 : -0"8 = 2 TO
I
(lO)
Lfdq)q 2 dq
where 0% is the conductivity of bulk metal (o 0
=
8~m2e2r3ro..,3h 3 ) and q = cos 0.
L4
I.I.!lI ERS
Since the variations in the reduced resistivity Pl.pP~ = a . c ~ . p with the magnitude H.~ of the longitudinal magnetic tield have the same aspect as those related to ,o~,po = % a~ given by Kao '~, comparison of these theoretical curves with the experimental results will yield further information on the mean free path and m o m e n t u m of the conduction electrons in the metal. This method could be sensitive since P~wPg is divided by 5 when the magnetic tield is multiplied by 2 [for a.l 0 = 0.01 ),o 3. Conclusion The longitudinal magnetoresistance effect in thin polycrystalline fihns may be conveniently described by eqn. (8). This suggests a new method for investigating the electrical properties of thin metal tilms. I 2 3 4 5 6 7 8 9 10 11 12
K. I,. ( ' h o p r a . R, Suri and A. 17. l ' h a k o o r , J .4ppl. Phy,s.. 4,~ (1977) 538. R. Suri, A, P. T h a k o o r and K. L. Chopra,./, Appl. I~hw., 46 (1975) 2574. A. Lal and V. P. Duggal, 7hm Solidkilm.~, 14(1972) 373. D (.~ohnayo and ,I. 1,. Sacedon. 7hm Solid It~ms, 35 (1976) 143, ( . H. l.ing, 77~itl Solidl'71ms. 16 (1973) 199. \". P. Dugga[ and V, P. Nagpal, J. Appl. Phys,. 42 ( 1971 ) 45(IO E. tl, Sondheimer. Adv. Phys., 1 (1952) I A. F, Mayadas and M. Shatzkes. Phys. R¢'~,, Sect, B. l ( [ 970) 138~. Y. t t. Kao, Phw. Rev.. Sc¢t, A. I3,~'(19651 [412 R (J. Chambers, P,'oc. R, Soc. l.on~hm. Set A, 202 ( 1951)1 378 ('. R. "l ellier, 771in Solid I.'ihns, 51 (19781 311. C . R . Tel lier and A. J. Tosser, lhin ,golid 1"TIm.~,43 (1977) 261.
Printed in the Netherlands
L1
Letter
Longitudinal magnetoresistance of thin metal films ( . R. II.II,I,IER, A. J. IOSSI~R A N I ) ( ' . PI('HARD
Univer.~it~; de Nancy 1. l,ahoratoire d'E/ectronique, ('.0. 140. 54037 N a m y COdex t k)'ume / ( Recei~ ed October 16. 1978 : accepted Oct ober 31, 1978)
1. Introduction In recent years it has been realized that the Hall coefficient R m. of thin metallic films subjected to both magnetic and electric fields will be strongly moditied ~-~'when the contribution to R m of structural defects such as grain boundaries "~ ~' cannot be regarded as negligible: thus G o l m a y o and Sacedon 't and Duggal and Nagpal 6 have suggested that the marked departure from the classical Fuchs-Sondheimer (FS) theory ~ of the thickness dependence of the Hall coefficient in metal films is due to the polycrystalline structure of these films. A suitable expression for the resistivity Pvp of a thin polycrystalline film in the absence of a magnetic field has recently been proposed by Mayadas and Shatzkcs~: they have assumed that the grain boundaries can be represented by two types of randomly spaced planes perpendicular and parallel rcspectively to the electric field and that the electronic scattering on parallel boundaries is only specular so that the problem reduces to a one-dimensional problem. In the case of a transverse magnetic field H: we have to consider applying the Sondheimer model" to two electric field components /='~ and E~.: hence in the Mayadas Shatzkes (MS) model we have to take into account two types of grain boundaries perpendicular to E.~ and E~. and the problem becomes very difficult to solve. Howevcr, in the case of a polycrystalline film subjected (Fig. 1) to both a longitudinal magnetic field (H,., O, O) and an electric field (E~. O, O) we may regard the magnctic field H as only modifying the electron trajectories :.'~.~°: the electrons move in helical paths and will also be scattered by the planes representing the grain boundaries which are parallel to the electric field E x (Fig. 1): as in the MS model we may assume that the electronic scattering from these planes is completely specular and can be neglected.
2. Theoretical It has been shown previously that the simultaneous grain boundary and background scattering effects in the MS model, with the geometry of Fig. 2, can be described by an effective relaxation time ref r which is given by;" J t r~.ff- = T0
+ ~fk~,l
(la)
where ~o refers to the background relaxation time, kF to the magnitude of the Fermi
L2
I.EITIiRS
//_._/// T [
,,
....
f f - qzJl ff
/
r'
IZ . . . . . . .
B
.... °°°+ ..... ,
~
Ex ~
;
'' \ ' ............. 'o
Fig. I. t h e geometry of the thin lihn and the applied lields, r = r o is taken to be the point B, BC is the projection of the helical path of an electron thal has been scattered at point B and c/; is the angle between the projection of the velocity vector v in the (y, z) plane and a fixed direction in this plane {taken parallel to the Oy axis in this easel. Fig 2. The geometry of the grain boundary model.
wavcvector a n d kx to the x c o m p o n e n t of the wavevector. F o r an electron travelling at an angle 0 to the x axis the k., c o m p o n e n t is k., = kt: cos 0
( 1b)
The physical p a r a m e t e r :t is related to the grain b o u n d a r y reflection coefficient r, the average grain size % and the b a c k g r o u n d mean free path Io b y ; :t =
It)
r
a~ l
--
(2)
r
Furthermore it has been shown 11.12 that the total solution of the external surfaces and grain boundary effects can be simply obtained on the basis of the FS theory. The method applies also to the +'kinetic model" developed by Chambers 1'' which describes the external size effect in a thin film. In the absence of a magnetic field and in the case of diffuse scattering on external surfaces the deviationJ~(k) of the d i s t r i b u t i o n functionJ~)(k) of the electrons is .ll(k.r)=
-
m
-
l-exp
-
13)
where t is the speed Ivl of the electrons and r - r o is a vector parallel to v (i.e. to k) which is related to electrons passing t h r o u g h a point r in the m e t a l : ro c o r r e s p o n d s to a point on the external surface of the tilm. On t a k i n g the total drift current and integrating over the cross-sectional area S the tilm c o n d u c t i v i t y o v is given as usual, in the absence of grain boundaries, b v ~°
O'l- ----
--, - , . ' ~ oh3S
f
o
+..i (~I o d r ~'V
£f£ dS
:~ dO
o
~' d 0 s i n 0 c o s 2 0 x
Hence the thin polycrystalline tilm c o n d u c t i v i t y %p m a y bc rewritten from eqn. (4) by substituting ~ft for to:
LETTERS
L3
2earn2
r 3 ~1o dr
o,.~ = 7.~S
dS
,r
de
d0 sin
,,
0 cos20x {5)
Equation (4) is still applicable in the case of thin fihns subjectcd to a longitudinal magnetic field provided that J r - r o t is taken to be the distance measured along the spiral trajectory of the electrons ~". Chambers' o has cvaluated integral {4) by noting that when an electron traverses from r to ro its projection on the I)'. z} plane moves an angle ¢ around a circle of radius R {Fig. 1) whcrc R =-
tHl"
elt.~ s i n 0 =
R osin0
{6)
The geometrical evaluation of [ r - r 0 ] by Chambers 1° is also valid for polycrystalline films and the distance J r - r o l may be written, for the geometry of Fig. I, [r-r,,[ =
Rotl,(y,z,~.Ol
{7)
The introduction ofeqns. (l), (6) and {7) in eqn. (5) gives a general expression for the polycrystalline film conductivity in the presence of a longitudinal magnetic field H~ :
avP
--
2e2m2"r°
x
h'~S
,
z'' ,'~ll, z- dr dS cr d.s
[
{
-¢R°_l+
l-exp
,
dq~
dO sin0c°s20 ] +~[cos0[ 1 x {8)
~,,, ~
Icos0l
Numerical evaluations of ¢ and ofeqn. {8} are somewhat complicated even if Kao's" formulation o r e in terms ofz, 0 and 4~ is used. It should be noted that eqn. [8) satislies the physical requirement that in the limit ~ = 0{no grain boundary effect} it reduces to the Chambers TM formulation. In the case of polycrystalline films with a constant grain size ag (thickness independent) the MS expression for the conductivity is deducedU from the FS expression by substituting the grain boundary' conductivity o-~ for the bulk conductivity a 0` where as is the conductivity of an intinitely thick polycrystalline film ~. Hence the grain boundary conductivity' c is given by eqn. [8) when Rc~ and S have intinite values:
~,=
2 e"23m- 2 ~r ° f * rh.3d~, l ° f : d"e; )
( 'l"
)
x c°s20 0:rlO sin /
)
dO.+-,c:sO~_,
(9)
Equation (9) is in agreement with the expression previously derived by Mayadas and Shatzkes 8 : -0"8 = 2 TO
I
(lO)
Lfdq)q 2 dq
where 0% is the conductivity of bulk metal (o 0
=
8~m2e2r3ro..,3h 3 ) and q = cos 0.
L4
I.I.!lI ERS
Since the variations in the reduced resistivity Pl.pP~ = a . c ~ . p with the magnitude H.~ of the longitudinal magnetic tield have the same aspect as those related to ,o~,po = % a~ given by Kao '~, comparison of these theoretical curves with the experimental results will yield further information on the mean free path and m o m e n t u m of the conduction electrons in the metal. This method could be sensitive since P~wPg is divided by 5 when the magnetic tield is multiplied by 2 [for a.l 0 = 0.01 ),o 3. Conclusion The longitudinal magnetoresistance effect in thin polycrystalline fihns may be conveniently described by eqn. (8). This suggests a new method for investigating the electrical properties of thin metal tilms. I 2 3 4 5 6 7 8 9 10 11 12
K. I,. ( ' h o p r a . R, Suri and A. 17. l ' h a k o o r , J .4ppl. Phy,s.. 4,~ (1977) 538. R. Suri, A, P. T h a k o o r and K. L. Chopra,./, Appl. I~hw., 46 (1975) 2574. A. Lal and V. P. Duggal, 7hm Solidkilm.~, 14(1972) 373. D (.~ohnayo and ,I. 1,. Sacedon. 7hm Solid It~ms, 35 (1976) 143, ( . H. l.ing, 77~itl Solidl'71ms. 16 (1973) 199. \". P. Dugga[ and V, P. Nagpal, J. Appl. Phys,. 42 ( 1971 ) 45(IO E. tl, Sondheimer. Adv. Phys., 1 (1952) I A. F, Mayadas and M. Shatzkes. Phys. R¢'~,, Sect, B. l ( [ 970) 138~. Y. t t. Kao, Phw. Rev.. Sc¢t, A. I3,~'(19651 [412 R (J. Chambers, P,'oc. R, Soc. l.on~hm. Set A, 202 ( 1951)1 378 ('. R. "l ellier, 771in Solid I.'ihns, 51 (19781 311. C . R . Tel lier and A. J. Tosser, lhin ,golid 1"TIm.~,43 (1977) 261.