APPLIED PHYSICS LETTERS
VOLUME 78, NUMBER 5
29 JANUARY 2001
Extraordinary magnetoresistance in externally shunted van der Pauw plates T. Zhou, D. R. Hines, and S. A. Solina) NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540
共Received 27 October 2000; accepted for publication 4 December 2000兲 We show that extraordinary magnetoresistance 共EMR兲 exhibited by a composite van der Pauw 共vdP兲 disk consisting of a semiconductor with an internal shunt can also be obtained from an electrically equivalent, externally shunted structure that is amenable to fabrication in the mesoscopic sizes required for important magnetic sensor applications. As an example, we use bilinear conformal mapping to transform the composite vdP disk into an externally shunted rectangular plate and calculate its EMR by solving Laplace’s equation with appropriate boundary conditions using no adjustable parameters. The calculations are in good agreement with measurements of InSb plates with Au shunts. Room-temperature EMR values as high as 550% at 0.05 T are obtained. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1343472兴 We have recently shown1,2 that room-temperature, extraordinary magnetoresistance 共EMR兲 as high as 100%, 9100%, and 750 000%, respectively, at magnetic fields of 0.05, 0.25 and 4.0 Tesla can be realized in a macroscopic 共dimensions ⬃1 mm兲 composite van der Pauw 共vdP兲 disk of a nonmagnetic, narrow-gap semiconductor into which is embedded a concentric cylindrical metallic inhomogeneity. The EMR obtained from such a centered vdP disk far exceeds that of magnetic materials such as those which exhibit giant MR3 or colossal MR4 and thus is of potential technological importance for use in magnetic sensors in a number of applications.5–7 Some of these applications, e.g., read heads in ultrahigh-density magnetic recording 共⬃1 Tb/in2兲, will require sensors of mesoscopic size 共dimensions ⬃300 Å兲. Because of the internal shunt, the centered vdP disk geometry will be very difficult to fabricate into mesoscopic structures of this size. In this letter we show that there is another class of structures which should not only be readily scalable to mesoscopic sizes but also yields even higher values of roomtemperature EMR than the centered vdP disk. We also show the mathematical relation between the centered vdP disk and the new structures and we show how to calculate the magnetotransport properties of the latter. It is known8 that any homogeneous device with a circular boundary of unit radius in the imaginary two-dimensional complex t plane with orthogonal axes r and is where t⫽r ⫹is can be mapped into the complex upper half Cartesian z plane with orthogonal axes x and iy where z⫽x⫹iy 关see Fig. 1共a兲兴 by using the bilinear transformation z(t)⫽⫺i 兵 关 t ⫹i兴/关t⫺i兴其.8 The above mapping equation transforms the four symmetrically spaced electrical contacts on the perimeter of the disk in the t-plane 关shown in Fig. 1共a兲 in the configuration for a magnetoresistance measurement兴 to the corresponding contacts on the line y⫽0 in the z plane. Although the mapped contacts are symmetric about the line x⫽0 they are not of equal size as they are when viewed in the t plane. If one embeds an off-centered hole of radius r 1 into the homogeneous disk of Fig. 1共a兲 as shown in Fig. 1共b兲, that hole maps to a line that truncates the upper half plane at a兲
Electronic mail:
[email protected]
height y 1 ⫽1/(r 1 ⫹1). In other words, the vacuum inside the hole of radius r 1 in the disk in the t plane maps to the vacuum above the line y 1 in the z plane. Consider now the circle of radius r 2 which creates an evacuated notch in the disk in the t plane as shown in Fig. 1共c兲. That circle maps to a line which truncates the z plane at the position x⫽x 2 ⫽1/r 2 as is also shown in Fig. 1共c兲. A symmetrically displaced circle of equal radius on the left of the vertical bisector of the disk in the t plane truncates the z plane with a line at position ⫺x 2 ⫽⫺(1/r 2 ) as shown in Fig. 1共d兲. By a selection of circular cuts in the t plane, the truncated disk can be exactly mapped to a rectangular structure of appropriate dimension in the z plane. Of the structures depicted in Figs. 1共a兲–1共d兲, that shown in Fig. 1共b兲 is the simplest one which contains a fully enclosed inhomogeneity, e.g., a circular hole displaced from the center of the disk. If we embed this hole with a highly conducting metal, then the resultant structure which we call an off-center vdP disk is similar to the centered vdP disk which yielded the large EMR values cited above.1,2 However, the corresponding rectangular mapped structure in the z plane would be of infinite extent in the ⫹x and ⫺x directions and would contain an external shunt of infinite height in the ⫹y direction. To avoid these complications, we define a structure which contains not only the r 2 cuts of Fig. 1共d兲 but also an additional circle of radius r 3 in the t plane as shown in Fig. 1共e兲. The latter maps to the line y⫽y 3 in the z plane. The modified off- centered vdP disk now contains a metallic inhomogeneity embedded into the space between the circles of radii r 1 , r 2 , and r 3 while the space between the circle of radius r 1 and the disk perimeter contains a narrow-gap semiconductor. Thus, the t-plane disk with an INTERNAL embedded shunt maps to a rectangle in the z plane with a corresponding EXTERNAL metallic shunt. Moreover, for the exact mapping depicted in Fig. 1共e兲, the electrical behavior of the two structures will be identical.8 Although the mapping technique described above has been known,8 the adaptation of this technique to the design of rectangular structures with external shunts has not been previously considered. Furthermore, for mapped plates with x 2 ⬎4, the cuts represented by the circles of radius r 2 in the
0003-6951/2001/78(5)/667/3/$18.00 667 © 2001 American Institute of Physics Downloaded 18 Sep 2007 to 132.163.130.151. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
668
Appl. Phys. Lett., Vol. 78, No. 5, 29 January 2001
Zhou, Hines, and Solin
FIG. 2. The parameters which define the geometry of a mapped off-center van der Pauw plate. In the diagram the superscripts m and s represent metal and semiconductor, respectively. The symbols J, E, and V represent, respectively, the current density, electric field, and voltage. The parameters a and b represent linear dimensions.
FIG. 1. Schematic diagram of bilinear mapping of 共a兲 a van der Pauw disk in the t plane mapped to the upper half space of the z plane. Also shown is the contact configuration for measuring magnetoresistance. 共b兲 An off-center hole in the t plane mapped to a line in the z plane, 共c兲 a circular perimeter cut in the t plane mapped to a line in the z plane, 共d兲 repeat of 共c兲 with a symmetric perimeter cut, 共e兲 an internal shunt in the t plane mapped to an external shunt in the z plane. Note: panels 共a兲–共d兲 follow Ref. 9, p. 163.
left panel of Fig. 1共e兲 are small/negligible. Therefore, the externally shunted plate structure shown on the right panel of Fig. 1共e兲 is, to a good approximation, electrically equivalent to the vdP disk shown in the left panel of Fig. 1共e兲 without the r 2 cuts. The calculation of the electrical properties of the structures of Fig. 1共e兲 is, however, more straightforward in the rectangular coordinates of the z plane than in the circular coordinates of the t plane. To illustrate this we show the solution method for the mapped rectangular plate of Fig. 1共e兲. The construct shown in Fig. 2 defines the parameters of the calculation of the EMR for a mapped externally shunted rectangular plate which is assumed to be of uniform thickness h. The length of the device is 2a, and the widths of the semiconductor and shunt are b and b 1 , respectively. The two current electrodes are placed on the outside, with distances to the centerline of a 1 and a 4 , and widths of ⌬a 1 and ⌬a 4 , respectively. The two voltage electrodes are treated as point contacts of zero width and are placed between the current contacts with distances to the centerline of a 2 and a 3 . We define  ⫽ H,  0 ⫽ 0 H, where and 0 are the mobilities of the semiconductor and metal, respectively, and H is the applied magnetic field normal to the plane of the device. We also note that the conductivity of the metal, 0⫽4.52 ⫻107 /⍀ m for Au is much larger than ⫽1.36⫻104 /⍀ m, the conductivity of the semiconductor. If we assume a constant current and no accumulated charge in the device, then the Maxwell equation reduce to the Laplace equation 2 V/ x 2 ⫹ 2 V/ y 2 ⫽0. We also assume that on the periphery of the device the current density is uniform at the two current leads, and zero
everywhere else; this defines the boundary conditions E tangential—continuous along the metal-semiconductor interface and J normal—continuous across the interface. Finally, we consider only the geometric contribution8 to the MR and ignore the physical contributions from the field dependence of the intrinsic properties of the semiconductor, e.g., and are assumed to be H-field independent. This is a good approximation for the low field regime we are considering here. We solve this problem analytically with NO adjustable parameters, and obtain the electrical potential V on the bottom edge of the device 共e.g., along the x axis in the z plane兲 as a function of  共or equivalently as a function of H兲, the dimensions of the device, and the position on the x axis of the z plane. Taking note that 0 Ⰷ , 0 Ⰶ , and setting b 1 Ⰷb we obtain V 共 兵 P 其 ,  ,x 兲 ⫽V 0 ⫹
共 1⫹  2 兲 2Ia h⌬a 1 2
⬁
⫻
1
兺 2 n⫽1 n
冉
A n cos
冊
nx nx , ⫺B n sin 2a 2a
共1兲
where V 0 is a constant, 兵 P 其 ⫽a 1 ,⌬a 1 ,a 2 ,a 3 ,a 4 , ⌬a 4 ,a,b,b 1 is the geometry parameter set A n⫽
S n J n ⫹  S 2n K n 1⫹  2 S 2n
S n ⫽tanh
冋
J n ⫽ sin
冋
B n⫽
1⫹  2 S 2n
, 共2兲
nb , 2a
n n n a ⫺sin 共 a ⫹⌬a 1 兲 ⫺sin 共 a ⫹⌬a 4 兲 2a 1 2a 1 2a 4
⫹sin and
,
S n K n ⫺  S 2n J n
册
n a , 2a 4
K n ⫽ ⫺cos
共3兲
n n n a 1 ⫹cos 共 a 1 ⫹⌬a 1 兲 ⫹cos 共n 2a 2a 2a 4
⫹⌬a 4 兲 ⫺cos
册
n a . 2a 4
共4兲
Downloaded 18 Sep 2007 to 132.163.130.151. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
Appl. Phys. Lett., Vol. 78, No. 5, 29 January 2001
Zhou, Hines, and Solin
669
Using the dependence of the electrical potential on the applied magnetic field given in Eq. 共1兲 above, one can calculate the EMR of the rectangular plate as a function of the specified parameters where EMR( 兵 P 其 ,  )⫽ 关 R( 兵 P 其 ,  ) ⫺R min(兵P其,)兴/Rmin(兵P,其), R( 兵 P 其 ,  )⫽⌬V( 兵 P 其 ,  )/I is the field and geometry-dependent effective resistance, I is a constant applied current, ⌬V is the output voltage, and R min(兵P其,) is the minimum value of the effective resistance which may be offset from H⫽0 if the placement of the voltage electrodes is asymmetric.9 For the symmetric electrode configuration, R min(兵P其,)⫽R(兵P其,0)⫽R 0 . In the case b 1 Ⰷb, it can be shown that the filling factor1 for the mapped plate is
␣⬅
a4 1⫹&
冉
冑
⫻ b⫹
1⫹
a4 1⫹&
2a 4 共 1⫹& 兲 b 1
冊册
.
冒 冋冉
1⫹
a4 共 1⫹& 兲 b 1
冊
共5兲
To test the above calculation and demonstrate that externally shunted rectangular structures can yield EMR of large magnitude, we fabricated a series of macroscopic devices of the type depicted schematically in Fig. 2. These devices were prepared from a 1.3 m active layer of metalorganic vapor phase epitaxy-grown epilayer of Te-doped n-type InSb 共electron concentration n⫽2.11⫻1022 m⫺3 and mobility ⫽4.02 m2/Vs兲 on a 4 inch semi-insulating GaAs wafer and passivated by a 200 nm layer of Si3N4. Details of the fabrication methods are provided elsewhere.1 The wafers were photolithographically patterned into chips bearing rectangular mesas with lateral dimensions of order 1 mm. The mesa contact pads and external shunt were simultaneously metallized with a Ti/Pt/Au stack, with Au the dominant component. Electrical contact to the electrodes of the devices was achieved by wire bonding. The results of the calculation described above in Eqs. 共1兲–共4兲 for both an asymmetric electrode configuration 共a 1 ⫽a 4 and a 2 ⫽a 3 兲 and a symmetric electrode configuration (a 1 ⫽a 4 and a 2 ⫽a 3 ) are shown in Fig. 3 panels 共a兲 and 共b兲, respectively, as solid lines for different plate geometries 共see the caption of Fig. 3 for parameter specifications兲. These calculations are compared in Fig. 3 with experimental measurements 共䊊, ⵜ兲 of the corresponding plates prepared in the manner described above. In view of the fact that there are no adjustable parameters in the calculation and that the effect of the physical MR is not included in the calculations but exists in the measurements, the agreement between theory and experiment is reasonable. Clearly, one can achieve significant room-temperature EMR with externally shunted rectangular plates. Indeed, the EMR of 550% obtained at 0.05 T for the plate with b/2a⫽1/21 共open circles in Fig. 3兲 and asymmetric voltage contacts is a factor of 5 higher than the corresponding maximum for the centered vdP disk with symmetric contacts which we previously studied.1 It is also evident from the data shown in Fig. 3 that the EMR of a mapped rectangular plate is dependent upon the size, shape, and location of the current and voltage contacts. This behavior is consistent with an analysis of the solutions
FIG. 3. A comparison of the theoretical solution 共solid lines兲 for the EMR of a mapped off-center vdP plate obtained using Eqs. 共1兲–共4兲 of the text with the experimental EMR of the corresponding structure for two sets of structures with the following parameters defined in Fig. 2: 共a兲 asymmetric voltage contacts 兵a 1 ⫽0.75 mm, ⌬a 1 ⫽0.01 mm, a 2 ⫽0 mm, a 3 ⫽0.35 mm, a 4 ⫽0.75 mm, ⌬a 4 ⫽0.01 mm, a⫽1.07 mm, b 1 ⫽0.9 mm其, 共b兲 symmetric voltage contacts 兵a 1 ⫽0.75 mm, ⌬a 1 ⫽0.01 mm, a 2 ⫽0.35 mm, a 3 ⫽0.35 mm a 4 ⫽0.75 mm, ⌬a 4 ⫽0.01 mm, a⫽1.07 mm, b 1 ⫽0.9 mm其. For both panels the filling factor has been calculated from Eq. 共5兲. With b ⫽0.1 mm, ␣ ⫽12/16⫺O while for b⫽0.3 mm, ␣ ⫽8/16⫺ⵜ.
to Laplace’s equation with various boundary conditions. The EMR is also dependent upon the length-to-width ratio of the semiconductor plate which determines the filling factor ␣ 关see Eq. 共5兲兴. The method for optimizing these parameters to achieve a given MR, output voltage, power signal to noise ratio, etc., will be addressed elsewhere.10 The authors thank M. W. Pelczynski and S. Schwed of EMCORE Corp. for providing the epilayer InSb samples and for assistance with sample fabrication. The authors also thank Tieke Thio and L. R. Ram-Mohan for useful discussions. 1
S. A. Solin, T. Thio, D. R. Hines, and J. J. Heremans, Science 289, 1530 共2000兲. 2 S. A. Solin, T. Thio, D. R. Hines, and J. J. Heremans, Proceedings of the 25th International Conference on the Physics of Semiconductors, Osaka, Japan, 2000. 3 W. F. Egelhoff, Jr., T. Ha, R. D. K. Misra, Y. Kadmon, J. Nir, C. J. Powell, M. D. Stiles, R. D. McMichael, C.-L. Lin, J. M. Sivertsen, J. H. Judy, K. Takano, A. E. Berkowitz, T. C. Anthony, and J. A. Brug, J. Appl. Phys. 78, 273 共1995兲. 4 S. Jin, M. McCormack, T. H. Tiefel, and R. Ramesh, J. Appl. Phys. 76, 6929 共1994兲. 5 J. P. Heremans, Mater. Res. Soc. Symp. Proc. 475, 63 共1997兲. 6 N. Kuze and I. Shibasaki, III–V’s Review 10, 28 共1997兲. 7 J. A. Brug, T. C. Anthony, and J. H. Nickel, MRS Bull. 21, 23 共1996兲. 8 R. S. Popovic, Hall Effect Devices 共Hilger, Bristol, 1991兲. 9 The determination of the actual sheet resistance of a van der Pauw disk in general requires two voltage measurements from distinct electrode pairs. But the operational performance of a sensor is characterized by the change of the effective resistance with magnetic field so we define the EMR in terms of the effective resistance. In the special case of fourfold symmetric contacts shown in Fig. 1共a兲, the effective and actual resistances are identical and only one voltage measurement is required. 10 T. Zhou, D. R. Hines, and S. A. Solin 共unpublished兲.
Downloaded 18 Sep 2007 to 132.163.130.151. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
