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Piezoresistance characteristics of some magnetic and non-magnetic metal films
Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
Piezoresistance characteristics of some magnetic and non-magnetic metal films S.U. Jen*, T.C. Wu, C.H. Liu Institute of Physics, Academia Sinica, 11529 Nankang, Taipei, Taiwan 11529, ROC Received 22 February 2002; received in revised form 6 May 2002
Abstract We have measured the strain–resistance characteristics of some non-magnetic and magnetic metal films. In order to suit to the objective of this article, we have chosen the Pd film as an example for the non-magnetic film, the Fe21Ni79 film for the magnetic film with a positive magnetostriction (ls > 0), and the Co20Ni80 film for the magnetic film with ( The strain gauge factor g of the Pd film was obtained from ls o0: The thickness tf of the films ranged from 40 to 1500 A. the linear strain–resistance plot. Then, g is expressed as a function of the sheet resistivity Rsq : For the magnetic films, only when the film is saturated either longitudinally (8) or transversely (>) by an external field, is the strain–resistance plot linear. While in a zero-field, the strain–resistance plot becomes non-linear. From the former, we could obtain g8 or g> ; and from the latter, we could obtain gap : g8 or g> is of the same order of magnitude as g: However, gap is abnormally large, due to the magnetoelastic and the anisotropic magnetoresistance effects. In addition to the piezoresistance data, we also found the critical thickness (tfc ) for the electrical coalescence and that (tfm ) for the magnetic coalescence. Usually, for the same magnetic material, tfm > tfc : The coalescence ability is the best for the Co20Ni80 films, and the worst for the Pd films. r 2002 Elsevier Science B.V. All rights reserved. PACS: 73.50.h; 75.80.+q Keywords: Piezoresistance; Magnetic properties; Non-magnetic properties; Metal films
1. Introduction In the past, the piezoresistance effect of some metals, either in a bulk or in a thinfilm form, has been studied [1,2]. The main purpose was to find out the strain gauge factor g; defined as dR0 dr ; ð1Þ ¼ ð1 þ 2nÞ þ g rDe R0 De *Corresponding author. Fax: +886-227834187. E-mail address: [email protected] (S.U. Jen).
where dR0 ¼ R R0 ; where R is the resistance of the sample, when stressed by a non-zero strain De; R0 the unstrained resistance, n the Poisson’s ratio, and dr=ðrDeÞ the strain coefficient of electrical resistivity. In addition, various models have been proposed to explain the gauge factor data observed. Recently, in particular, people were interested in the gauge factor enhancement phenomena in some homogenous or heterogeneous films, which were prepared near the coalescence condition [3]. Although the g values of critically coalesced films might be more scattered than that
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 3 7 1 - 2
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
of continuous films, they were correct, especially for those non-magnetic films [2]. However, due to the domain-rotation and the anisotropic magnetoresistance (AMR) effects, earlier data for the magnetic films were incomplete and should be discussed again, though not necessarily using the same materials [2]. The point is that, as mentioned in Ref. [4], the apparent gauge factor gap of a magnetic film, such as the Ni film, can be quite large and of a magnitude about 20. The discussion about gap in Ref. [4] was very brief, and the film sample used by them was assumed to be continuous. More discussion is needed. In this article, we shall give detailed studies on the piezoresistance characteristics of some magnetic and non-magnetic metal or alloy films. Especially, attentions will be paid on those films near their coalescence regions.
2. Experiments Patterned (a 7-mm long and 2-mm wide sample with the leads aside) magnetic and non-magnetic films were made by the evaporation method in an 1 106 Torr vacuum. The substrate was a piece of 0211 Corning glass. The substrate temperature was TS ¼ 1801C. Here, both the Fe21Ni79 and the Co20Ni80 alloy films are considered magnetic, while the Pd film is considered non-magnetic. In
55
order to induce an easy axis (EA), a deposition field of h ¼ 4800 A/m was applied along the width of the film sample. After deposition, the substrate was cut into an isosceles triangle, so that the sample was along the symmetry line of the triangle and one end of the sample was about 4 mm away from the base, as shown in Fig. 1. The film thickness tf was measured again by a step profiler, and the average thickness ts of substrate was determined from the density method. To form a horizontal cantilever, as shown in Fig. 1, the base was champed and the opposite vertex was free. A force F ; either downward or upward, was applied to the free end to induce a longitudinal tensile (De > 0) or compressive (Deo0) strain on the film sample. The reason to choose the isosceles triangle shape is because the strain induced along the length of the sample is uniform [5], i.e. De is independent of the longitudinal distance (x) from the base. Next, in order to find the relationship between F and De; we replaced the film sample by a strain gauge. If ts is held fixed, the force F versus strain De plot is linear, i.e. F ¼ kDe; where k is a constant, as shown in Fig. 2. Further, from elastic theory, one can show that k is proportional to ðts Þ2 : For the commercially available 0211 glass substrate, its thickness ts usually ranges from 140 to 220 mm. Hence, we did a series of calibrations on the substrates with different ts : Parts of the results
Fig. 1. Experimental setup for the piezoresistance measurement used in this article. The triangular shape substrate is clamped at base to form a cantilever. The shaded area represents the film sample.
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respectively. More details will be discussed in Section 3. In regard to the auxiliary measurements, there were two kinds. One was to measure the saturation magnetostriction ls of the film sample by employing the optical-cantilever method [7], and the other was to measure the anisotropy field HK from the hard-axis (HA) hysteresis loop by using the vibrating sample magnetometer (VSM).
3. Piezoresistance of magnetic films
Fig. 2. Calibration curve of a blank 0211 glass substrate. ts is the thickness of the substrate. F is the force applied at the free end of the cantilever. De is the measured strain.
are illustrated in Fig. 2. We find that k=ðts Þ2 ¼ 1:005 mN/mm2. For the piezoresistance (or the strain–resistance) measurement, since F and ts were known, the corresponding strain De on the film sample could be calculated from the previous calibration curve. As to the electrical resistance measurement, there were two kinds: if R0 of the sample is higher than 1 kO, the two-wire method was used, and if R0 is lower, the four-wire method was used [6]. Note, to avoid the DC drifting problem of the digital multimeter (Keithley 2000), the electric response or the change in resistance R0 was recorded almost simultaneously, when the strain De was applied. In general, the external De was increased from 250 106 to 250 106. However, for the magnetic film samples, besides the zero-field piezoresistance measurement, described above, two similar measurements were carried out; one with the longitudinal field H8 and the other with the transverse field H> ; respectively. Further, in order to saturate the sample in either direction, the magnitude of the fixed field is at least 4800 A/m. As mentioned in Section 1, for the non-magnetic sample, there will be only one strain gauge factor, i.e. g; but for the magnetic sample, there will be four, i.e. gap ; g0 ; g8 ; and g> ; which corresponds to the zero-field, the H8 ; and the H> cases,
For non-magnetic metal films, some possible mechanisms for the observed piezoresistance behavior were discussed [1–3]. In short, if the film is continuous (including the bulk), the free electron model is utilized to explain the tf dependence of g; and if the film is near the coalescence region, the tunneling conduction mechanism is used to account for the high value of g: In order to define the meaning of coalescence more consistently (or quantitatively), we take the following criteria: (1) the sheet resistivity Rsq is defined as Rsq ¼ R0 ðw=LÞ; where w and L are width and probe length of the film sample, (2) if Rsq o104 O, the film is considered to be continuous, (3) if 107 O >Rsq > 104 O; the film is critically coalesced, and (4) if Rsq > 107 O, the film is discontinuous. Note the definitions above are just for the convenience of comparison. The word ‘‘continuous’’ means that most of the grains are physically or electrically connected to each other with few voids inside the grain boundary. Therefore, the other word ‘‘discontinuous’’ means that the electron must tunnel through the gaps among grains. However, in later discussion, we shall introduce another term about continuous, i.e. how the magnetization Ms (or the flux line) inside the grain is magnetically shorted to its neighbor’s? Briefly speaking, the strain–resistance curve of a non-magnetic film, such as the Pd film, should be linear, as shown in Fig. 3. By a linear fit using Eq. (1), we find the value of g: For magnetic metal films, the situations of the zero-field strain–resistance curve are more complicated. In general, two types of strain–resistance behavior should be observed. Fig. 4(a) is for the
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
Fig. 3. The strain–resistance plot for the Pd and the Co20Ni80 (with H> ¼ 4800 A/m) films. tf is the thickness of the film. dR0 ¼ R2R0 :
ls > 0 film, such as the Fe21Ni79 film. Due to the pure straining and the AMR effects, the net strain– resistance plot should look like curve BAOCDE in Fig. 4(a). We shall explain this result in details. Because ls De > 0; from the magnetoelastic effect, if De is tensile and small, Ms in each domain gradually rotates from the (as-deposited) EA direction, also the transverse (>) direction, toward the (as-deposited) HA direction, also the longitudinal (8) direction [8]. Due to the AMR effect, R ¼ R0 þ DR sin2 y and R8 > R>, where DR ¼ R8 2R> and y is the angle between the magnetization and the transverse directions, R should increase from R0 as represented by the section OC of Fig. 4(a). From Ref. [8], if the tensile strain continue to increase and reaches a critical value Dec ; defined as Dec
Fig. 4. The analysis of zero-field strain–resistance plot for the magnetic metal films: (a) in the ls > 0 case and (b) in the ls o0 case. ls is the saturation magnetostriction.
57
2Ku ; 3ls Ef
ð2Þ
where Ku is the uniaxial anisotropy energy and Ef is the Young’s modulus of the Fe21Ni19 film, Ms switches abruptly toward the longitudinal (8) axis through the domain-wall rotation and/or translation. Therefore, R increases greatly at De ¼ Dec > 0: In other words, beyond section CD the ‘‘new’’ EA is defined along the axis of the tensile stress. Hence, when De > Deð8; SÞ; Ms is exactly parallel or anti-parallel to the longitudinal direction. Then, the ‘‘pure’’ tensile strain effect on R8 will be observed. As a result, the section DE should be linear and the slope is equal to g8 : Alternately, if Deo0; the magnetoelastic and the AMR effects will lead to the dotted line OH. The reason is as follows. From domain observation, when R ¼ R0 ; we have the closure domains. Hence, the negative De will cause R to decrease from R0 to R> : Beyond point H by the compressive strain, i.e. De % Deð>; SÞo0; the AMR part in R> is in principle independent of De: That is the slope of the dotted line HI is zero. Nevertheless, at the same time, the pure compressive strain will affect R> and result in the dotted line OFG. Therefore, the net (or observed) strain–resistance curve should look like section OAB. Note the slope of section AB is equal to g> : Secondly, Fig. 4(b) is for the ls o0 film, such as the Co20Ni80 film. The net strain–resistance should look like EDCOAB. The
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S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
explanation is similar, except the following two features are different: (A) Dec o0 in Fig. 4(b), and (B) the slopes of lines OH and OF in Fig. 4(a) always have the same sign, while that in Fig. 4(b) may have the opposite sign. In practice, the maximum slope of curve C at De ¼ Dec in Fig. 4 is not infinite. Therefore, as discussed later, for the Fe21Ni79 film we may estimate the slope and define g0 ðrightÞ gap ; and for the Co20Ni80 film g0 ðleftÞ gap from Fig. 4, respectively. Further, in Fig. 4(a) or (b), the transition points D and A are usually not easy to define, because of the demagnetizing fields near the edges and in the corners of the sample. Hence, instead of using Fig. 4 to obtain g8 and g> ; we used the strain–resistance curves under the condition of the saturation field, H8 and H> ; respectively. As shown in Fig. 3, when H> ¼ 4800 A/m, the strain–resistance curve is linear again. From the hysteresis loops, the coercive force Hc of Fe21Ni79 is about 80 A/m, and that of Co20Ni80 is about 592 A/m. In addition, HK p384 A/m for Fe21Ni79, and HK p2720 A/m for Co20Ni80. A field with the magnitude of 4800 A/m is large enough to saturate the film in either direction. g> and g8 so obtained should be more reliable.
4. Results and discussion Fig. 5 shows the zero-strain resistivity r of Pd film as a function of tf : As in Refs. [9,10], r can be expressed as r ¼ a þ ðb=tf Þ; where a and b are constants. From Figs. 5 and 6, the coalescence ( This stage is approached, when tf ¼ tfc ¼ 47 A. indicates that the Pd film adheres well on the glass substrate.1 Besides, its ability to form a continuous film is also good [11]. The Rsq dependence of the gauge factor g for the Pd film is shown in Fig. 6. All the three features about g; as mentioned in Ref. [2], are also observed in Fig. 6: (i) for low Rsq (thicker films) the g values tend to approach the bulk value gb ¼ 4:06; (ii) there is a minimum in g at Rsq ¼ 490 O, and (iii) for high Rsq (thinner films) g increases greatly. Using our previous criteria, we 1 When compared with adhesion of the Au or the Cu film to glass substrate, that of the Pd film is much better.
find that when Pd films are continuous, 1:89pgp3:00; and when they are critically coalesced, g reaches 21–26. In Fig. 7, we have shown the result of the strain– resistance plot in the zero-field condition for the ( Fe21Ni79 film. Because ls of Fe21Ni79 tf ¼ 200 A film is negative, from discussion in Section 3, the plot in Fig. 7 is non-linear and looks similar to Fig. 4(a). Then, Table 1 summarizes all the relevant features in Figs. 4(a) and 7. Those include g0 (left), gap ; g8 ; g> ; r; Rsq ; Dec ; ls ; and HK of the Fe21Ni79 films as a function of tf : g0 (left) means the slope of line OA in Fig. 4(a). Hence, from Fig. 7 the point A locates around De ¼ 125 106 : When tf is held fixed, we see that g0 (left) is larger than either g8 or g> : The reason has been explained in Section 3. Although the value of g8 is close to that g> ; they are not exactly the same to each other. We may take either g> or g8 and plot it as a function of Rsq : The result is similar to that of Fig. 6. That means when a magnetic film has been saturated, its piezoresistance behavior is in general like that of a non-magnetic film. In other words, g8 and g> are of the same order of magnitude as g: From the r versus tf plot or the g> (or g8 ) versus Rsq plot (Tables 1 and 2), it is easy to tell that when ( for Fe21Ni79 and/or tf ¼ tfc ¼ 41 A ( tf ¼ tfc ¼ 45 A for Co20Ni80, the film starts to coalesce, at least in the morphological or the electrical sense. Therefore, we conclude that from the electrical point of view the coalescence ability on glass is the best for the Co20Ni80 film. From Table 1 and Fig. 7, it is found that g0 ðrightÞ gap is abnormally large. The reason has been explained in last section. Especially, it agrees with the theory that for the Fe21Ni79 film both Dec > 0 and ls > 0: Moreover, it can be easily verified that for each tf case, Eq. (2) is satisfied, where Ku ¼ 12 Is HK ; and Ef ¼ 2:0 1011 N/m2. In addition, we find that from Table 1, gap decreases as tf decreases. This is probably due to the fact that both Ms and AMR are decreasing functions for decreasing tf : Then, another interesting phe( the anomaly nomenon is observed: when tf p50 A, of gap disappears and the strain–resistance relation becomes linear. For more details, in Table 1, when the non-linear anomaly sometimes appears and sometimes does not, we note ‘‘yes/no’’ under the
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
59
Fig. 5. The zero-strain resistivity r of the Pd film plotted as a function of the film thickness tf :
Fig. 6. The strain gauge factor g of the Pd film plotted as a function of the sheet resistivity Rsq :
g0 (right) column, and when it is linear definitely, we note ‘‘no’’. Hence, it is concluded that when ( the flux of the magnetic path (or tf ptfm ¼ 50 A, circuit) through the Fe21Ni79 film is greatly interrupted, when crossing the gaps, and the AMR coupling between the electric and the magnetic entities is essentially zero there. Therefore, we may view tfm as the critical film thickness for the magnetic coalescence. In Fig. 8, the result of the strain–resistance plot ( in the H ¼ 0 condition is shown for the tf ¼ 100 A Co20Ni80 film. Clearly, Fig. 8 is also non-linear
and looks like Fig. 4(b). All the relevant features are summarized in Table 2 for the Co20Ni80 films. g0 (right) means the slope of line OA in Fig. 4(b). As an example, from Fig. 8, the point A should locate around De ¼ 75 106 : However, as discussed in Section 3, because the slopes of lines OH and OF in Fig. 4(b) have the opposite sign, g0 (right) is smaller than either g8 or g> in general. ( film is In particular, g0 (right) of the tf ¼ 100 A even negative (e.g. Table 2 and Fig. 8). As to tfm of ( from Table 2. the Co20Ni80 film, it is equal to 45 A Finally, Table 2 and Fig. 8 show that both Deo0
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
60
Fig. 7. The zero-field strain–resistance curve of the Fe21Ni79 film.
Table 1 The various piezoresistance coefficients (or gauge factors) g0 ; g8 ; g> ; electrical resistivity r; sheet resistivity Rsq ; critical strain Dec ; anisotropy field HK ; and magnetostriction ls of the Fe21Ni79 Permalloy films ( tf (A)
g0 (left)
g0 (right)a
g8
g>
r (mO cm)
Rsq (O)
Dec (106)
ls (106)
HK (A/m)
1500 1000 500 200 100 80 50 48 45
2.91 2.90 3.09 2.19 1.80 0.90 2.65 2.61 3.63
300 150 30 60 72 — Yes/no No No
— 1.87 1.28 1.86 — 1.08 1.34 1.64 3.16
— 1.81 1.21 1.89 — 0.97 1.37 1.55 3.44
31.83 35.16 42.10 40.80 45.60 1.01 102 1.23 102 1.94 102 1.96 104
2.12 3.52 8.42 20.40 45.60 1.27 102 2.45 102 3.88 102 4.37 104
114 105 119 141 47 — 34 — —
4.12 4.51 5.50 3.67 4.42 — 1.83 — —
352 — 384 — 304 — 240 — —
a
g0 ðrightÞ ¼ gap :
Table 2 The various piezoresistance coefficients (or gauge factors) g0 ; g8 ; g> ; electrical resistivity r; sheet resistivity Rsq ; critical strain Dec ; anisotropy field HK ; and magnetostriction ls of the Co20Ni80 films ( tf (A)
g0 (left)a
g0 (right)
g8
g>
r (mO cm)
Rsq (O)
Dec (106)
ls (106)
HK (A/m)
1500 800 200 100 50 45 41
320 — — 62 55 Yes/no No
4.64 1.91 0.65 1.44 0.46 0.22 3.08
— 1.81 1.71 1.71 1.70 1.35 3.06
2.23 2.07 1.71 1.60 1.99 1.35 3.23
20.91 23.17 34.21 49.19 59.39 1.12 102 1.91 105
1.39 2.90 17.10 49.19 1.19 102 2.48 102 4.79 105
202 200 202 121 160 102 —
28.5 27.9 26.2 22.4 17.5 — —
2720 — 1960 1200 1360 — —
a
g0 ðleftÞ ¼ gap :
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
61
Fig. 8. The zero-field strain–resistance curve of the Co20Ni80 film.
and ls o0 for the Co20Ni80 film. Once again, these results agree with Eq. (2). Obviously, for the magnetic metal films, either the Fe21Ni79 or the Co20Ni80 film, tfm does not equal to tfc in general. Also, we find that tfm > tfc :
5. Conclusion Non-magnetic metal, such as Pd, film and magnetic metal, such as Fe21Ni79 and Co20Ni80, films were made. The thickness tf ranged from 40 ( The piezoresistance (or strain–resisto 1500 A. tance) characteristics of these film samples were measured. The strain gauge factor g of the Pd film was obtained as a function of Rsq from the linear strain–resistance plot. However, for the magnetic films, besides g0 (left) and g0 (right) from the nonlinear zero-field plot, we obtained g8 and g> from the linear saturation-field plot. For the Fe21Ni79 film, g0 ðrightÞ ¼ gap ; and for the Co20Ni80 film, g0 ðleftÞ ¼ gap : gap is abnormally large due to the magnetoelastic and the AMR effects. Because ls > 0 for the Fe21Ni79 film, gap occurs in the positive De (tensile) region, i.e. Dec > 0: For the same reason, since ls o0 for the Co20Ni80 film, gap occurs in the negative De (compressive) region, i.e. Dec o0: Further, we are able to roughly identify the location of the saturation point A in the zero-field strain–resistance curve of the Fe21Ni79 or Co20Ni80 film. g8 or g of the magnetic films is about the same
order of magnitude as g of the non-magnetic film. With regard to the coalescence of these films on the glass substrate, we find that besides the critical thickness (tfc ) for the electrical coalescence in the non-magnetic and the magnetic metal films, there exists another critical thickness (tfm ) for the magnetic coalescence in the magnetic film. In general, tfm > tfc in the same film material. From this study, it is found that tfc of the Pd, Fe21Ni79 ( respecand Co20Ni80 films are 47, 45, and 41 A, tively, and tfm of the Fe21Ni79 and Co20Ni80 films ( respectively. Hence, we conclude are 50 and 45 A, that among the three kinds of films being tested, the Co20Ni80 film exhibits the best coalescence ability and the Pd film the worst.
Acknowledgements We are thankful to the National Science Council for the financial support of this work.
References [1] G.C. Kuczynski, Phys. Rev. 94 (1954) 61. [2] R.L. Parker, A. Krinsky, J. Appl. Phys. 34 (1963) 2700. [3] C. Grimaldi, P. Ryser, S. Str.assler, J. Appl. Phys. 90 (2001) 322. [4] T.R. McGuire, R.I. Potter, IEEE Trans. MAG-11 (1975) 1018. [5] R.M. McClintock, Rev. Sci. Instr. 30 (1959) 715.
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[6] H.H. Wieder, in: C. Laird (Ed.), Laboratory Notes on Electrical and Galvanomagnetic Measurements, Elsevier, Amsterdam, 1979. [7] S.U. Jen, H.P. Chiang, C.M. Chung, M.N. Kao, J. Magn. Magn. Mater. 236 (2001) 312.
[8] J.D. Livingston, Phys. Stat. Sol. 70A (1982) 591. [9] H. Hoffmann, J. Vancea, Thin Solid Films 85 (1981) 147. [10] S.U. Jen, C.T. Wei, J.Y. Huang, J. Magn. Magn. Mater. 164 (1996) 49. [11] S. Nakahara, Thin Solid Films 45 (1977) 421.
Piezoresistance characteristics of some magnetic and non-magnetic metal films S.U. Jen*, T.C. Wu, C.H. Liu Institute of Physics, Academia Sinica, 11529 Nankang, Taipei, Taiwan 11529, ROC Received 22 February 2002; received in revised form 6 May 2002
Abstract We have measured the strain–resistance characteristics of some non-magnetic and magnetic metal films. In order to suit to the objective of this article, we have chosen the Pd film as an example for the non-magnetic film, the Fe21Ni79 film for the magnetic film with a positive magnetostriction (ls > 0), and the Co20Ni80 film for the magnetic film with ( The strain gauge factor g of the Pd film was obtained from ls o0: The thickness tf of the films ranged from 40 to 1500 A. the linear strain–resistance plot. Then, g is expressed as a function of the sheet resistivity Rsq : For the magnetic films, only when the film is saturated either longitudinally (8) or transversely (>) by an external field, is the strain–resistance plot linear. While in a zero-field, the strain–resistance plot becomes non-linear. From the former, we could obtain g8 or g> ; and from the latter, we could obtain gap : g8 or g> is of the same order of magnitude as g: However, gap is abnormally large, due to the magnetoelastic and the anisotropic magnetoresistance effects. In addition to the piezoresistance data, we also found the critical thickness (tfc ) for the electrical coalescence and that (tfm ) for the magnetic coalescence. Usually, for the same magnetic material, tfm > tfc : The coalescence ability is the best for the Co20Ni80 films, and the worst for the Pd films. r 2002 Elsevier Science B.V. All rights reserved. PACS: 73.50.h; 75.80.+q Keywords: Piezoresistance; Magnetic properties; Non-magnetic properties; Metal films
1. Introduction In the past, the piezoresistance effect of some metals, either in a bulk or in a thinfilm form, has been studied [1,2]. The main purpose was to find out the strain gauge factor g; defined as dR0 dr ; ð1Þ ¼ ð1 þ 2nÞ þ g rDe R0 De *Corresponding author. Fax: +886-227834187. E-mail address: [email protected] (S.U. Jen).
where dR0 ¼ R R0 ; where R is the resistance of the sample, when stressed by a non-zero strain De; R0 the unstrained resistance, n the Poisson’s ratio, and dr=ðrDeÞ the strain coefficient of electrical resistivity. In addition, various models have been proposed to explain the gauge factor data observed. Recently, in particular, people were interested in the gauge factor enhancement phenomena in some homogenous or heterogeneous films, which were prepared near the coalescence condition [3]. Although the g values of critically coalesced films might be more scattered than that
0304-8853/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 3 7 1 - 2
S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
of continuous films, they were correct, especially for those non-magnetic films [2]. However, due to the domain-rotation and the anisotropic magnetoresistance (AMR) effects, earlier data for the magnetic films were incomplete and should be discussed again, though not necessarily using the same materials [2]. The point is that, as mentioned in Ref. [4], the apparent gauge factor gap of a magnetic film, such as the Ni film, can be quite large and of a magnitude about 20. The discussion about gap in Ref. [4] was very brief, and the film sample used by them was assumed to be continuous. More discussion is needed. In this article, we shall give detailed studies on the piezoresistance characteristics of some magnetic and non-magnetic metal or alloy films. Especially, attentions will be paid on those films near their coalescence regions.
2. Experiments Patterned (a 7-mm long and 2-mm wide sample with the leads aside) magnetic and non-magnetic films were made by the evaporation method in an 1 106 Torr vacuum. The substrate was a piece of 0211 Corning glass. The substrate temperature was TS ¼ 1801C. Here, both the Fe21Ni79 and the Co20Ni80 alloy films are considered magnetic, while the Pd film is considered non-magnetic. In
55
order to induce an easy axis (EA), a deposition field of h ¼ 4800 A/m was applied along the width of the film sample. After deposition, the substrate was cut into an isosceles triangle, so that the sample was along the symmetry line of the triangle and one end of the sample was about 4 mm away from the base, as shown in Fig. 1. The film thickness tf was measured again by a step profiler, and the average thickness ts of substrate was determined from the density method. To form a horizontal cantilever, as shown in Fig. 1, the base was champed and the opposite vertex was free. A force F ; either downward or upward, was applied to the free end to induce a longitudinal tensile (De > 0) or compressive (Deo0) strain on the film sample. The reason to choose the isosceles triangle shape is because the strain induced along the length of the sample is uniform [5], i.e. De is independent of the longitudinal distance (x) from the base. Next, in order to find the relationship between F and De; we replaced the film sample by a strain gauge. If ts is held fixed, the force F versus strain De plot is linear, i.e. F ¼ kDe; where k is a constant, as shown in Fig. 2. Further, from elastic theory, one can show that k is proportional to ðts Þ2 : For the commercially available 0211 glass substrate, its thickness ts usually ranges from 140 to 220 mm. Hence, we did a series of calibrations on the substrates with different ts : Parts of the results
Fig. 1. Experimental setup for the piezoresistance measurement used in this article. The triangular shape substrate is clamped at base to form a cantilever. The shaded area represents the film sample.
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S.U. Jen et al. / Journal of Magnetism and Magnetic Materials 256 (2003) 54–62
respectively. More details will be discussed in Section 3. In regard to the auxiliary measurements, there were two kinds. One was to measure the saturation magnetostriction ls of the film sample by employing the optical-cantilever method [7], and the other was to measure the anisotropy field HK from the hard-axis (HA) hysteresis loop by using the vibrating sample magnetometer (VSM).
3. Piezoresistance of magnetic films
Fig. 2. Calibration curve of a blank 0211 glass substrate. ts is the thickness of the substrate. F is the force applied at the free end of the cantilever. De is the measured strain.
are illustrated in Fig. 2. We find that k=ðts Þ2 ¼ 1:005 mN/mm2. For the piezoresistance (or the strain–resistance) measurement, since F and ts were known, the corresponding strain De on the film sample could be calculated from the previous calibration curve. As to the electrical resistance measurement, there were two kinds: if R0 of the sample is higher than 1 kO, the two-wire method was used, and if R0 is lower, the four-wire method was used [6]. Note, to avoid the DC drifting problem of the digital multimeter (Keithley 2000), the electric response or the change in resistance R0 was recorded almost simultaneously, when the strain De was applied. In general, the external De was increased from 250 106 to 250 106. However, for the magnetic film samples, besides the zero-field piezoresistance measurement, described above, two similar measurements were carried out; one with the longitudinal field H8 and the other with the transverse field H> ; respectively. Further, in order to saturate the sample in either direction, the magnitude of the fixed field is at least 4800 A/m. As mentioned in Section 1, for the non-magnetic sample, there will be only one strain gauge factor, i.e. g; but for the magnetic sample, there will be four, i.e. gap ; g0 ; g8 ; and g> ; which corresponds to the zero-field, the H8 ; and the H> cases,
For non-magnetic metal films, some possible mechanisms for the observed piezoresistance behavior were discussed [1–3]. In short, if the film is continuous (including the bulk), the free electron model is utilized to explain the tf dependence of g; and if the film is near the coalescence region, the tunneling conduction mechanism is used to account for the high value of g: In order to define the meaning of coalescence more consistently (or quantitatively), we take the following criteria: (1) the sheet resistivity Rsq is defined as Rsq ¼ R0 ðw=LÞ; where w and L are width and probe length of the film sample, (2) if Rsq o104 O, the film is considered to be continuous, (3) if 107 O >Rsq > 104 O; the film is critically coalesced, and (4) if Rsq > 107 O, the film is discontinuous. Note the definitions above are just for the convenience of comparison. The word ‘‘continuous’’ means that most of the grains are physically or electrically connected to each other with few voids inside the grain boundary. Therefore, the other word ‘‘discontinuous’’ means that the electron must tunnel through the gaps among grains. However, in later discussion, we shall introduce another term about continuous, i.e. how the magnetization Ms (or the flux line) inside the grain is magnetically shorted to its neighbor’s? Briefly speaking, the strain–resistance curve of a non-magnetic film, such as the Pd film, should be linear, as shown in Fig. 3. By a linear fit using Eq. (1), we find the value of g: For magnetic metal films, the situations of the zero-field strain–resistance curve are more complicated. In general, two types of strain–resistance behavior should be observed. Fig. 4(a) is for the
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Fig. 3. The strain–resistance plot for the Pd and the Co20Ni80 (with H> ¼ 4800 A/m) films. tf is the thickness of the film. dR0 ¼ R2R0 :
ls > 0 film, such as the Fe21Ni79 film. Due to the pure straining and the AMR effects, the net strain– resistance plot should look like curve BAOCDE in Fig. 4(a). We shall explain this result in details. Because ls De > 0; from the magnetoelastic effect, if De is tensile and small, Ms in each domain gradually rotates from the (as-deposited) EA direction, also the transverse (>) direction, toward the (as-deposited) HA direction, also the longitudinal (8) direction [8]. Due to the AMR effect, R ¼ R0 þ DR sin2 y and R8 > R>, where DR ¼ R8 2R> and y is the angle between the magnetization and the transverse directions, R should increase from R0 as represented by the section OC of Fig. 4(a). From Ref. [8], if the tensile strain continue to increase and reaches a critical value Dec ; defined as Dec
Fig. 4. The analysis of zero-field strain–resistance plot for the magnetic metal films: (a) in the ls > 0 case and (b) in the ls o0 case. ls is the saturation magnetostriction.
57
2Ku ; 3ls Ef
ð2Þ
where Ku is the uniaxial anisotropy energy and Ef is the Young’s modulus of the Fe21Ni19 film, Ms switches abruptly toward the longitudinal (8) axis through the domain-wall rotation and/or translation. Therefore, R increases greatly at De ¼ Dec > 0: In other words, beyond section CD the ‘‘new’’ EA is defined along the axis of the tensile stress. Hence, when De > Deð8; SÞ; Ms is exactly parallel or anti-parallel to the longitudinal direction. Then, the ‘‘pure’’ tensile strain effect on R8 will be observed. As a result, the section DE should be linear and the slope is equal to g8 : Alternately, if Deo0; the magnetoelastic and the AMR effects will lead to the dotted line OH. The reason is as follows. From domain observation, when R ¼ R0 ; we have the closure domains. Hence, the negative De will cause R to decrease from R0 to R> : Beyond point H by the compressive strain, i.e. De % Deð>; SÞo0; the AMR part in R> is in principle independent of De: That is the slope of the dotted line HI is zero. Nevertheless, at the same time, the pure compressive strain will affect R> and result in the dotted line OFG. Therefore, the net (or observed) strain–resistance curve should look like section OAB. Note the slope of section AB is equal to g> : Secondly, Fig. 4(b) is for the ls o0 film, such as the Co20Ni80 film. The net strain–resistance should look like EDCOAB. The
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explanation is similar, except the following two features are different: (A) Dec o0 in Fig. 4(b), and (B) the slopes of lines OH and OF in Fig. 4(a) always have the same sign, while that in Fig. 4(b) may have the opposite sign. In practice, the maximum slope of curve C at De ¼ Dec in Fig. 4 is not infinite. Therefore, as discussed later, for the Fe21Ni79 film we may estimate the slope and define g0 ðrightÞ gap ; and for the Co20Ni80 film g0 ðleftÞ gap from Fig. 4, respectively. Further, in Fig. 4(a) or (b), the transition points D and A are usually not easy to define, because of the demagnetizing fields near the edges and in the corners of the sample. Hence, instead of using Fig. 4 to obtain g8 and g> ; we used the strain–resistance curves under the condition of the saturation field, H8 and H> ; respectively. As shown in Fig. 3, when H> ¼ 4800 A/m, the strain–resistance curve is linear again. From the hysteresis loops, the coercive force Hc of Fe21Ni79 is about 80 A/m, and that of Co20Ni80 is about 592 A/m. In addition, HK p384 A/m for Fe21Ni79, and HK p2720 A/m for Co20Ni80. A field with the magnitude of 4800 A/m is large enough to saturate the film in either direction. g> and g8 so obtained should be more reliable.
4. Results and discussion Fig. 5 shows the zero-strain resistivity r of Pd film as a function of tf : As in Refs. [9,10], r can be expressed as r ¼ a þ ðb=tf Þ; where a and b are constants. From Figs. 5 and 6, the coalescence ( This stage is approached, when tf ¼ tfc ¼ 47 A. indicates that the Pd film adheres well on the glass substrate.1 Besides, its ability to form a continuous film is also good [11]. The Rsq dependence of the gauge factor g for the Pd film is shown in Fig. 6. All the three features about g; as mentioned in Ref. [2], are also observed in Fig. 6: (i) for low Rsq (thicker films) the g values tend to approach the bulk value gb ¼ 4:06; (ii) there is a minimum in g at Rsq ¼ 490 O, and (iii) for high Rsq (thinner films) g increases greatly. Using our previous criteria, we 1 When compared with adhesion of the Au or the Cu film to glass substrate, that of the Pd film is much better.
find that when Pd films are continuous, 1:89pgp3:00; and when they are critically coalesced, g reaches 21–26. In Fig. 7, we have shown the result of the strain– resistance plot in the zero-field condition for the ( Fe21Ni79 film. Because ls of Fe21Ni79 tf ¼ 200 A film is negative, from discussion in Section 3, the plot in Fig. 7 is non-linear and looks similar to Fig. 4(a). Then, Table 1 summarizes all the relevant features in Figs. 4(a) and 7. Those include g0 (left), gap ; g8 ; g> ; r; Rsq ; Dec ; ls ; and HK of the Fe21Ni79 films as a function of tf : g0 (left) means the slope of line OA in Fig. 4(a). Hence, from Fig. 7 the point A locates around De ¼ 125 106 : When tf is held fixed, we see that g0 (left) is larger than either g8 or g> : The reason has been explained in Section 3. Although the value of g8 is close to that g> ; they are not exactly the same to each other. We may take either g> or g8 and plot it as a function of Rsq : The result is similar to that of Fig. 6. That means when a magnetic film has been saturated, its piezoresistance behavior is in general like that of a non-magnetic film. In other words, g8 and g> are of the same order of magnitude as g: From the r versus tf plot or the g> (or g8 ) versus Rsq plot (Tables 1 and 2), it is easy to tell that when ( for Fe21Ni79 and/or tf ¼ tfc ¼ 41 A ( tf ¼ tfc ¼ 45 A for Co20Ni80, the film starts to coalesce, at least in the morphological or the electrical sense. Therefore, we conclude that from the electrical point of view the coalescence ability on glass is the best for the Co20Ni80 film. From Table 1 and Fig. 7, it is found that g0 ðrightÞ gap is abnormally large. The reason has been explained in last section. Especially, it agrees with the theory that for the Fe21Ni79 film both Dec > 0 and ls > 0: Moreover, it can be easily verified that for each tf case, Eq. (2) is satisfied, where Ku ¼ 12 Is HK ; and Ef ¼ 2:0 1011 N/m2. In addition, we find that from Table 1, gap decreases as tf decreases. This is probably due to the fact that both Ms and AMR are decreasing functions for decreasing tf : Then, another interesting phe( the anomaly nomenon is observed: when tf p50 A, of gap disappears and the strain–resistance relation becomes linear. For more details, in Table 1, when the non-linear anomaly sometimes appears and sometimes does not, we note ‘‘yes/no’’ under the
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Fig. 5. The zero-strain resistivity r of the Pd film plotted as a function of the film thickness tf :
Fig. 6. The strain gauge factor g of the Pd film plotted as a function of the sheet resistivity Rsq :
g0 (right) column, and when it is linear definitely, we note ‘‘no’’. Hence, it is concluded that when ( the flux of the magnetic path (or tf ptfm ¼ 50 A, circuit) through the Fe21Ni79 film is greatly interrupted, when crossing the gaps, and the AMR coupling between the electric and the magnetic entities is essentially zero there. Therefore, we may view tfm as the critical film thickness for the magnetic coalescence. In Fig. 8, the result of the strain–resistance plot ( in the H ¼ 0 condition is shown for the tf ¼ 100 A Co20Ni80 film. Clearly, Fig. 8 is also non-linear
and looks like Fig. 4(b). All the relevant features are summarized in Table 2 for the Co20Ni80 films. g0 (right) means the slope of line OA in Fig. 4(b). As an example, from Fig. 8, the point A should locate around De ¼ 75 106 : However, as discussed in Section 3, because the slopes of lines OH and OF in Fig. 4(b) have the opposite sign, g0 (right) is smaller than either g8 or g> in general. ( film is In particular, g0 (right) of the tf ¼ 100 A even negative (e.g. Table 2 and Fig. 8). As to tfm of ( from Table 2. the Co20Ni80 film, it is equal to 45 A Finally, Table 2 and Fig. 8 show that both Deo0
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Fig. 7. The zero-field strain–resistance curve of the Fe21Ni79 film.
Table 1 The various piezoresistance coefficients (or gauge factors) g0 ; g8 ; g> ; electrical resistivity r; sheet resistivity Rsq ; critical strain Dec ; anisotropy field HK ; and magnetostriction ls of the Fe21Ni79 Permalloy films ( tf (A)
g0 (left)
g0 (right)a
g8
g>
r (mO cm)
Rsq (O)
Dec (106)
ls (106)
HK (A/m)
1500 1000 500 200 100 80 50 48 45
2.91 2.90 3.09 2.19 1.80 0.90 2.65 2.61 3.63
300 150 30 60 72 — Yes/no No No
— 1.87 1.28 1.86 — 1.08 1.34 1.64 3.16
— 1.81 1.21 1.89 — 0.97 1.37 1.55 3.44
31.83 35.16 42.10 40.80 45.60 1.01 102 1.23 102 1.94 102 1.96 104
2.12 3.52 8.42 20.40 45.60 1.27 102 2.45 102 3.88 102 4.37 104
114 105 119 141 47 — 34 — —
4.12 4.51 5.50 3.67 4.42 — 1.83 — —
352 — 384 — 304 — 240 — —
a
g0 ðrightÞ ¼ gap :
Table 2 The various piezoresistance coefficients (or gauge factors) g0 ; g8 ; g> ; electrical resistivity r; sheet resistivity Rsq ; critical strain Dec ; anisotropy field HK ; and magnetostriction ls of the Co20Ni80 films ( tf (A)
g0 (left)a
g0 (right)
g8
g>
r (mO cm)
Rsq (O)
Dec (106)
ls (106)
HK (A/m)
1500 800 200 100 50 45 41
320 — — 62 55 Yes/no No
4.64 1.91 0.65 1.44 0.46 0.22 3.08
— 1.81 1.71 1.71 1.70 1.35 3.06
2.23 2.07 1.71 1.60 1.99 1.35 3.23
20.91 23.17 34.21 49.19 59.39 1.12 102 1.91 105
1.39 2.90 17.10 49.19 1.19 102 2.48 102 4.79 105
202 200 202 121 160 102 —
28.5 27.9 26.2 22.4 17.5 — —
2720 — 1960 1200 1360 — —
a
g0 ðleftÞ ¼ gap :
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Fig. 8. The zero-field strain–resistance curve of the Co20Ni80 film.
and ls o0 for the Co20Ni80 film. Once again, these results agree with Eq. (2). Obviously, for the magnetic metal films, either the Fe21Ni79 or the Co20Ni80 film, tfm does not equal to tfc in general. Also, we find that tfm > tfc :
5. Conclusion Non-magnetic metal, such as Pd, film and magnetic metal, such as Fe21Ni79 and Co20Ni80, films were made. The thickness tf ranged from 40 ( The piezoresistance (or strain–resisto 1500 A. tance) characteristics of these film samples were measured. The strain gauge factor g of the Pd film was obtained as a function of Rsq from the linear strain–resistance plot. However, for the magnetic films, besides g0 (left) and g0 (right) from the nonlinear zero-field plot, we obtained g8 and g> from the linear saturation-field plot. For the Fe21Ni79 film, g0 ðrightÞ ¼ gap ; and for the Co20Ni80 film, g0 ðleftÞ ¼ gap : gap is abnormally large due to the magnetoelastic and the AMR effects. Because ls > 0 for the Fe21Ni79 film, gap occurs in the positive De (tensile) region, i.e. Dec > 0: For the same reason, since ls o0 for the Co20Ni80 film, gap occurs in the negative De (compressive) region, i.e. Dec o0: Further, we are able to roughly identify the location of the saturation point A in the zero-field strain–resistance curve of the Fe21Ni79 or Co20Ni80 film. g8 or g of the magnetic films is about the same
order of magnitude as g of the non-magnetic film. With regard to the coalescence of these films on the glass substrate, we find that besides the critical thickness (tfc ) for the electrical coalescence in the non-magnetic and the magnetic metal films, there exists another critical thickness (tfm ) for the magnetic coalescence in the magnetic film. In general, tfm > tfc in the same film material. From this study, it is found that tfc of the Pd, Fe21Ni79 ( respecand Co20Ni80 films are 47, 45, and 41 A, tively, and tfm of the Fe21Ni79 and Co20Ni80 films ( respectively. Hence, we conclude are 50 and 45 A, that among the three kinds of films being tested, the Co20Ni80 film exhibits the best coalescence ability and the Pd film the worst.
Acknowledgements We are thankful to the National Science Council for the financial support of this work.
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