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Mechanical Derivation of the Longitudinal and Transverse Piezoresistive Coefficient on Piezoresistive Pressure Sensor
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 29 000–000 (2012) 1612 – 1617 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE)
Mechanical Derivation of the Longitudinal and Transverse Piezoresistive Coefficient on Piezoresistive Pressure Sensor Guang Yanga*, Hengyan Xiea a
College of Engineering, Heilongjiang Bayi Agricultural University, 2 Xin Yang Road , Development Zone, Daqing163319, China
Abstract Piezoresistive pressure sensors widely used in industrial production is an important member of the microelectromechanical systems (MEMS) sensors. In order to make good use of the pressure sensors they are made up of some force-sensitive components from single-crystal silicon. This paper presented the coordinate transformation relation of piezoresistive coefficient matrix based on elasticity. Afterwards, Bond’s transformation matrix of stress connected with Bond’s transformation matrix of strain to deal with piezoresistive coefficients matrix after the coordinate transformation can simplify calculations. Using symbolic Computation of MATLAB program, we derived the longitudinal piezoresistive coefficient
π L and
transverse piezoresistive coefficient
π T that
accorded with the
results of general method, which proves the derivation. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology
Keywords: Piezoresistive pressure sensors;Coordinate transformation relation; Bond matrix method; longitudinal and transverse piezoresistive coefficient
1. Introduction In 1856, Load Kelvin first found the piezoresistive effect in iron and copper [1].The so-called piezoresistive effect denotes that the resistances value change when strain and deformation occur in resistances. On the study of Piezoresistive characteristics of single-crystal silicon, Smith [2] indicated that no other than π 44 , π 55 , π 66 remained in all the shear piezoresistive coefficient if single-crystal silicon * Corresponding author. Tel.:+0086-459-6819216. E-mail address: [email protected].
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.182
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Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
2
was cubic crystal system, and
π 44 = π 55 = π 66 . Mason and Thurston [3], according to the relationship of
the electric field strength and current density, piezoresistive coefficient and stress, got the general formula of longitudinal piezoresistive coefficient π L and transverse piezoresistive coefficient π T .
π L = π11 − 2(π11 − π12 − π 44 )(l12 m12 + m12 n12 + l12 n12 )
(1)
π T = π 12 + (π 11 − π 12 − π 44 )(l12l22 + m12 m22 + n12 n22 )
(2) This paper gets the conversion relationships of the piezoresistive coefficient matrix in the old and new coordinates according to elasticity and simplifies the matrix by connecting Bond stress transformation matrix with Bond and strain transformation matrix [4,5],then computes the longitudinal π L and transverse
πT
piezoresistive coefficient by the Matlabmatrix notation calculation, which is very
meaningful to understand and apply the principles of piezoresistive pressure sensor. 2 Piezoresistive coefficient matrix of cubic single-crystal silicon In the cubic single-crystal silicon to take on a micro-unit shown in Figure 1, there are six independent stress components σ 11 , σ 22 , σ 33 , σ 23 = σ 32 , σ 31 = σ 13 , σ 12 = σ 21 .
3 σ
33
σ
32
σ
31
σ
13
σ
21
ο σ
σ
23
σ
22
2
σ
12
11
1 Fig.1. Micro-unit and stress components are shown.
For the anisotropic single crystal silicon material, if 1, 2 and 3 axis are along the <1 0 0> crystal axis, the relationships of the relative changes components in resistivity (or resistance) of the cubic singlecrystal silicon and stress components [1], which is transformed by 11 → 1, 22 → 2, 33 → 3, 23=32 → 4, 13=31 → 5, 12=21 → 6, will be expressed as ⎧ δ1 ⎫ ⎪δ ⎪ ⎪ 2⎪ ⎪⎪δ 3 ⎪⎪ = ⎨ ⎬ ⎪δ 4 ⎪ ⎪δ 5 ⎪ ⎪ ⎪ ⎪⎩δ 5 ⎪⎭
⎡ Δρ11 / ρ ⎤ ⎢ ⎥ ⎢ Δρ 22 / ρ ⎥ ⎢ Δρ33 / ρ ⎥ ⎢= ⎥ ⎢ Δρ 23 / ρ ⎥ ⎢ Δρ31 / ρ ⎥ ⎢ ⎥ ⎢⎣ Δρ12 / ρ ⎥⎦
⎡ ΔR11 / R ⎤ ⎢ ⎥ ⎢ ΔR22 / R ⎥ ⎢ ΔR33 / R ⎥ ⎢= ⎥ ⎢ ΔR23 / R ⎥ ⎢ ΔR31 / R ⎥ ⎢ ⎥ ⎢⎣ ΔR12 / R ⎥⎦
0 0 ⎤ ⎧σ 1 ⎫ ⎡π 11 π 12 π 12 0 ⎢π 0 0 ⎥⎥ ⎪⎪σ 2 ⎪⎪ ⎢ 12 π 11 π 12 0 ⎢π 12 π 12 π 11 0 0 0 ⎥ ⎪⎪σ 3 ⎪⎪ , ⎢ ⎥⎨ ⎬ 0 0 π 44 0 0 ⎥ ⎪σ 4 ⎪ ⎢0 ⎢0 0 0 0 π 44 0 ⎥ ⎪σ 5 ⎪ ⎢ ⎥⎪ ⎪ 0 0 0 0 π 44 ⎥⎦ ⎪⎩σ 6 ⎭⎪ ⎢⎣ 0
(3)
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where
Guang YangH. and XieEngineering / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, XieHengyan / Procedia 00 (2011) 000–000
δ i are
3
the relative change components in resistivity or resistance, σ i are stress components,
Δρij are the change components in resistivity, and ΔRij are the change components in resistance,
π 11 is
longitudinal piezoresistive coefficient,which indicates that the stress along a crystal axis will create effect of the relative changes components of in resistivity or resistance along the crystal axis, π 12 is transverse piezoresistive coefficient,which indicates that the stress along a crystal axis will create effect of the relative changes components in resistivity or resistance along the perpendicular crystal axis, π 44 is shear piezoresistive coefficient, which indicates that the shear stress will create effect of the relative changes components of in resistivity or resistance along the shear stress direction. The matrix of (3) is (4) {δ } = [π ]{σ } 3 Coordinate transformation expression of cubic crystal silicon piezoresistive coefficient matrix Four resistors (Fig.2 shows a piezoresistive pressure sensor) on the silicon diaphragm is of equivalent resistances R1 , R2 , R3 , R4 , which are located in four-side midpoints of the silicon diaphragm at maximum stress. The four resistors in circuit make up Wheatstone full bridge circuit and such changes of four resistors can be shown by voltage changes.
Fig.2.Piezoresistive pressure sensor[1].
In order to understand conveniently, we don’t transform the subscript temporarily. In Fig.3, the resistors R2 , R3 of sensor, current I and longitudinal stress σ L are along 1' axis; the resistors R1 , R4 of '
sensor and transverse σ T are along 2 axis. At the same time, the four resistors are located in O 1' 2 ' plane. We assume that the stress components σ 11 , σ 22 , σ 33 , σ 23 = σ 32 , σ 31 = σ 13 σ 12 = σ 21 of micro-unit at arbitrary point O are known. In O − 123 coordinates, the plane ABC connecting with coordinate plane by the point O is made up of tetrahedron OABC . We suppose that the outside normal of the plane ABC is along '
the axis 1 of the new coordinate system and the axis 2' , 3' are determined by the right-hand. Direction cosines are expressed as cos( 1' ,1)= l1 ,cos( 1' ,2)= m1 ,cos( 1' ,3)= n1 , cos( 2' ,1)= l2 ,cos( 2' ,2)=
m2 ,cos( 2' ,3)= n2 ,and cos( 3' ,1)= l3 ,cos( 3' ,2）= m3 ,cos( 3' ,3）= n3 .The matrix of direction cosines is an orthogonal matrix.
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Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
4
3
σ
2'
21
R1
,
11
σ
σ
p
12
22
1
,
R3
I
2
p σ
σο
R2
p
3
σ
L
R4
σ
1'
σ
T
13
2
σ
23
31
σ
32
1
3'
σ
33
Fig.3.Resistance position and stress of silicon diaphragm in the new coordinate system are expressed.
The stress relations are obtained by Cauchy stress relation and stress axis formula between old and new coordinates, which is expressed as ⎧ σ 11' ' ⎫ ⎡ l12 m12 n12 2m1n1 2l1n1 2l1m1 ⎤ ⎧σ 11 ⎫ ⎪σ ⎪ ⎢ 2 ⎥⎪ ⎪ 2 2 m2 n2 2m2 n2 2l2 n2 2l2 m2 ⎥ ⎪σ 22 ⎪ ⎪ 2 ' 2 ' ⎪ ⎢ l2 (5) ⎪⎪σ 3'3' ⎪⎪ ⎢ l32 m32 n32 2m3 n3 2l3 n3 2l3 m3 ⎥ ⎪⎪σ 33 ⎪⎪ . ⎥⎨ ⎬ ⎨ ⎬=⎢ ⎪σ 2'3' ⎪ ⎢l2l3 m2 m3 n2 n3 m2 n3 + m3 n2 l2 n3 + l3n2 l2 m3 + l3 m2 ⎥ ⎪σ 23 ⎪ ⎪σ 31' ' ⎪ ⎢ l1l3 m1m3 n1n3 m1n3 + m3 n1 l1n3 + l3n1 l1m3 + l3 m1 ⎥ ⎪σ 31 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎩⎪σ 1'2' ⎭⎪ ⎣⎢ l1l2 m1m2 n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ ⎪⎩σ 12 ⎪⎭ Formula (5) is simplified as
{σ } = [ M ]{σ } ,
(6)
'
where [ M ] is Bond stress transformation matrix[4,5]. According to the same principle, we can find the relationships of the relative change in resistivity between old and new coordinates ⎧ δ11 ⎫ ⎡ l12 m12 n12 2m1n1 2l1n1 2l1m1 ⎤ ⎧δ11 ⎫ ⎪δ ⎪ ⎢ 2 ⎥⎪ ⎪ 2 2 l m n 2 m n 2 l n 2 l2 m2 ⎥ ⎪δ 22 ⎪ 2 2 2 2 2 2 ⎪ 22 ⎪ ⎢ 2 2 (7) m32 n32 2m3n3 2l3 n3 2l3m3 ⎥ ⎪⎪δ 33 ⎪⎪ , ⎪⎪δ 3 3 ⎪⎪ ⎢ l3 ⎥⎨ ⎬ ⎨ ⎬=⎢ ⎪δ 2 3 ⎪ ⎢l2l3 m2 m3 n2 n3 m2 n3 + m3n2 l2 n3 + l3 n2 l2 m3 + l3m2 ⎥ ⎪δ 23 ⎪ ⎪ δ 31 ⎪ ⎢ l1l3 m1m3 n1n3 m1n3 + m3n1 l1n3 + l3 n1 l1m3 + l3 m1 ⎥ ⎪δ 31 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎩⎪δ12 ⎭⎪ ⎣⎢ l1l2 m1m2 n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ ⎪⎩δ12 ⎪⎭ Formula (7) can be expressed as ' '
' ' ' '
' ' ' '
' '
{δ } = [ M ]{δ }
(8)
'
Substituting (6), (8) to (4), {δ
{δ } = [ M ][π ][ M ] {σ } , '
−1
'
'
} is got (9)
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Guang YangH. and Hengyan Xie Engineering / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, Xie / Procedia 00 (2011) 000–000
5
so the expression ⎡π ' ⎤ can be found ⎣ ⎦ −1 ⎡⎣π ' ⎤⎦ = [ M ][π ][ M ] .
(10)
Bond strain transformation matrix is ⎡ l12 m12 n12 m1n1 l1n1 l1m1 ⎤ ⎢ 2 ⎥ 2 2 m2 n2 m2 n2 l2 n2 l2 m2 ⎥ ⎢ l2 ⎢ l2 m32 n32 m3 n1 l3 n3 l3m3 ⎥ [N ] = ⎢ 3 ⎥ ⎢ 2l2l3 2m2 m3 2n2 n3 m2 n3 + m3n2 l2 n3 + l3n2 l2 m3 + l3m2 ⎥ ⎢ 2l l 2m m 2n n m n + m n l n + l n l m + l m ⎥ 1 3 1 3 1 3 3 1 1 3 3 1 1 3 3 1 ⎢ 13 ⎥ ⎢⎣ 2l1l2 2m1m2 2n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ By [ M ]−1 = [ N ]T ， ⎡π ' ⎤ is simplified
(11)
⎣ ⎦
⎡⎣π ' ⎤⎦ = [ M ][π ][ N ]
T
⎡ l12 ⎢ 2 ⎢ l2 ⎢ l2 =⎢ 3 ⎢l2l3 ⎢l l ⎢13 ⎢⎣ l1l2
m12
n12
2m1n1
2l1n1
m22 m32
n22 n32
2m2 n2 2m3 n3
2l2 n2 2l3n3
m2 m3 m1m3 m1m2
n2 n3 n1n3 n1n2
m2 n3 + m3 n2 m1n3 + m3 n1 m1n2 + m2 n1
l2 n3 + l3n2 l1n3 + l3n1 l1n2 + l2 n1
0 0 ⎤ ⎡ l12 ⎤ ⎡π 11 π 12 π 12 0 ⎢ ⎥⎢ 0 0 ⎥⎥ ⎢ l22 2l2 m2 ⎥ ⎢π 12 π 11 π 12 0 0 0 ⎥ ⎢ l32 2l3m3 ⎥ ⎢π 12 π 12 π 11 0 ⎥⎢ ⎥⎢ 0 0 π 44 0 0 ⎥ ⎢ 2l2l3 l2 m3 + l3m2 ⎥ ⎢ 0 ⎥ 0 0 0 π 44 0 ⎥ ⎢ 2l1l3 l1m3 + l3 m1 ⎢ 0 ⎥⎢ ⎥⎢ 0 0 0 0 π 44 ⎥⎦ ⎣⎢ 2l1l2 l1m2 + l2 m1 ⎦⎥ ⎢⎣ 0 2l1m1
m12
n12
m1n1
l1n1
m22 m32
n22 n32
m2 n2 m3 n1
l2 n2 l3n3
2m2 m3 2m1m3 2m1m2
2n2 n3 2n1n3 2n1n2
m2 n3 + m3n2 m1n3 + m3n1 m1n2 + m2 n1
l2 n3 + l3n2 l1n3 + l3n1 l1n2 + l2 n1
⎤ ⎥ l2 m2 ⎥ l3m3 ⎥ ⎥ l2 m3 + l3 m2 ⎥ l1m3 + l3m1 ⎥ ⎥ l1m2 + l2 m1 ⎦⎥ l1m1
T
(12)
4 The calculation of longitudinal piezoresistive coefficient π L and transverse piezoresistive coefficient π T Shown in Fig.2, the shape of silicon diaphragm on pressure sensor is a square. Under external pressure the bending deformation of the diaphragm is small and the thickness of resistance in comparision with the width and the length is negligible. So the force of the resistance can be approximated as a plane stress state σ 33 = σ 23 = σ 31 =0. Therefore, we only consider the longitudinal and transverse piezoresistive coefficient after coordinate transformation ( π L = π 11' , π T = π 12' ). Formula (14) using MATLAB matrix symbol evaluates is changed to transpi = [ (L1^2*p11+M1^2*p12+N1^2*p12)*conj(L1)^2+(L1^2*p12+M1^2*p11+N1^2*p12)*conj(M1)^2+(L1^2*p12+ M1^2*p12+N1^2*p11)*conj(N1)^2+2*M1*N1*p44*conj(M1*N1)+2*L1*N1*p44*conj(L1*N1)+2*M1*L1*p44*c onj(M1*L1), (L1^2*p11+M1^2*p12+N1^2*p12)*conj(L2)^2+(L1^2*p12+M1^2*p11+N1^2*p12)*conj(M2)^2+(L1^2*p12+M1^ 2*p12+N1^2*p11)*conj(N2)^2+2*M1*N1*p44*conj(M2*N2)+2*L1*N1*p44*conj(L2*N2)+2*M1*L1*p44*conj( M2*L2), …]
Here the author keeps the coordinate transformation calculations π 11 ， π 12 and omits other items. By '
the relationship l12 + m12 + n12 = 1 , the first item
'
π 11' of the calculation transpi is simplified to
2 2 2 2 2 2 π L = π 11' = π 11 − 2(π 11 − π 12 − π 44 )(l1 m1 + m1 n1 + l1 n1 )
By the relationship (l12 + m12 + n12 )(l22 + m22 + n22 ) = 1 and l1l2 + m1m2 + n1n2 = 0 , the second item the calculation transpi is simplified to π T = π 12' = π 12 + (π 11 − π 12 − π 44 )(l12l22 + m12 m22 + n12 n22 )
π 12' of
Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
6
If we know the longitudinal stress component σ L and the transverse stress component σ T , the relative change of resistance can be got by the output of voltage through the Wheatstone full bridge circuit, by the item ΔR ≈ π σ + π σ . R
L
L
T
T
5Conclusions Piezoresistive coefficient matrix belonging to fourth-order tensor is similar to the stiffness matrix and co-mpliance matrix. If coordinate transformation matrix of piezoresistive coefficient with tensor in the old and the new coordinate system is very abstract, we derived the coordinate transformation of piezoresistive coefficient matrix by using an intuitive mechanical methods and simplified the transformation matrix with the Bond strain and stress transformation matrix. We obtained the formulas of the longitudinal and tra-nsverse piezoresistive coefficient π L , π T after symbolic Computation of MATLAB , which is useful to understand how the relative change in resistance are shown through output of the Wheatstone full bridge circuit. Acknowledgements This research is funded by the Research Fund for the Doctoral Program of Higher Education of China (Grant NO. 20102305120003). References [1] C. Liu,Foundations of MEMS,North-America,New Jersey,2006. [2] C. S. Smith,Piezoresistance effect in germanium and silicon,Phys Rev94(1954)42–49. [3] W.P.Mason and R.N.Thurston,Use of piezoresistive materials in the measurement of displacement, force, and torque,J. Acous. Soc. of Am. 29(1957)1096–1101. [4] W.Bond, The mathematics of the physical properties of crystals,BSTJ 22(1943)1-72. [5] B.A. Auld,Acoustic fields and waves in solids volume,1973.
1617
Procedia Engineering
ProcediaProcedia Engineering 00 (2011) Engineering 29 000–000 (2012) 1612 – 1617 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE)
Mechanical Derivation of the Longitudinal and Transverse Piezoresistive Coefficient on Piezoresistive Pressure Sensor Guang Yanga*, Hengyan Xiea a
College of Engineering, Heilongjiang Bayi Agricultural University, 2 Xin Yang Road , Development Zone, Daqing163319, China
Abstract Piezoresistive pressure sensors widely used in industrial production is an important member of the microelectromechanical systems (MEMS) sensors. In order to make good use of the pressure sensors they are made up of some force-sensitive components from single-crystal silicon. This paper presented the coordinate transformation relation of piezoresistive coefficient matrix based on elasticity. Afterwards, Bond’s transformation matrix of stress connected with Bond’s transformation matrix of strain to deal with piezoresistive coefficients matrix after the coordinate transformation can simplify calculations. Using symbolic Computation of MATLAB program, we derived the longitudinal piezoresistive coefficient
π L and
transverse piezoresistive coefficient
π T that
accorded with the
results of general method, which proves the derivation. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology
Keywords: Piezoresistive pressure sensors;Coordinate transformation relation; Bond matrix method; longitudinal and transverse piezoresistive coefficient
1. Introduction In 1856, Load Kelvin first found the piezoresistive effect in iron and copper [1].The so-called piezoresistive effect denotes that the resistances value change when strain and deformation occur in resistances. On the study of Piezoresistive characteristics of single-crystal silicon, Smith [2] indicated that no other than π 44 , π 55 , π 66 remained in all the shear piezoresistive coefficient if single-crystal silicon * Corresponding author. Tel.:+0086-459-6819216. E-mail address: [email protected].
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.182
1613
Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
2
was cubic crystal system, and
π 44 = π 55 = π 66 . Mason and Thurston [3], according to the relationship of
the electric field strength and current density, piezoresistive coefficient and stress, got the general formula of longitudinal piezoresistive coefficient π L and transverse piezoresistive coefficient π T .
π L = π11 − 2(π11 − π12 − π 44 )(l12 m12 + m12 n12 + l12 n12 )
(1)
π T = π 12 + (π 11 − π 12 − π 44 )(l12l22 + m12 m22 + n12 n22 )
(2) This paper gets the conversion relationships of the piezoresistive coefficient matrix in the old and new coordinates according to elasticity and simplifies the matrix by connecting Bond stress transformation matrix with Bond and strain transformation matrix [4,5],then computes the longitudinal π L and transverse
πT
piezoresistive coefficient by the Matlabmatrix notation calculation, which is very
meaningful to understand and apply the principles of piezoresistive pressure sensor. 2 Piezoresistive coefficient matrix of cubic single-crystal silicon In the cubic single-crystal silicon to take on a micro-unit shown in Figure 1, there are six independent stress components σ 11 , σ 22 , σ 33 , σ 23 = σ 32 , σ 31 = σ 13 , σ 12 = σ 21 .
3 σ
33
σ
32
σ
31
σ
13
σ
21
ο σ
σ
23
σ
22
2
σ
12
11
1 Fig.1. Micro-unit and stress components are shown.
For the anisotropic single crystal silicon material, if 1, 2 and 3 axis are along the <1 0 0> crystal axis, the relationships of the relative changes components in resistivity (or resistance) of the cubic singlecrystal silicon and stress components [1], which is transformed by 11 → 1, 22 → 2, 33 → 3, 23=32 → 4, 13=31 → 5, 12=21 → 6, will be expressed as ⎧ δ1 ⎫ ⎪δ ⎪ ⎪ 2⎪ ⎪⎪δ 3 ⎪⎪ = ⎨ ⎬ ⎪δ 4 ⎪ ⎪δ 5 ⎪ ⎪ ⎪ ⎪⎩δ 5 ⎪⎭
⎡ Δρ11 / ρ ⎤ ⎢ ⎥ ⎢ Δρ 22 / ρ ⎥ ⎢ Δρ33 / ρ ⎥ ⎢= ⎥ ⎢ Δρ 23 / ρ ⎥ ⎢ Δρ31 / ρ ⎥ ⎢ ⎥ ⎢⎣ Δρ12 / ρ ⎥⎦
⎡ ΔR11 / R ⎤ ⎢ ⎥ ⎢ ΔR22 / R ⎥ ⎢ ΔR33 / R ⎥ ⎢= ⎥ ⎢ ΔR23 / R ⎥ ⎢ ΔR31 / R ⎥ ⎢ ⎥ ⎢⎣ ΔR12 / R ⎥⎦
0 0 ⎤ ⎧σ 1 ⎫ ⎡π 11 π 12 π 12 0 ⎢π 0 0 ⎥⎥ ⎪⎪σ 2 ⎪⎪ ⎢ 12 π 11 π 12 0 ⎢π 12 π 12 π 11 0 0 0 ⎥ ⎪⎪σ 3 ⎪⎪ , ⎢ ⎥⎨ ⎬ 0 0 π 44 0 0 ⎥ ⎪σ 4 ⎪ ⎢0 ⎢0 0 0 0 π 44 0 ⎥ ⎪σ 5 ⎪ ⎢ ⎥⎪ ⎪ 0 0 0 0 π 44 ⎥⎦ ⎪⎩σ 6 ⎭⎪ ⎢⎣ 0
(3)
1614
where
Guang YangH. and XieEngineering / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, XieHengyan / Procedia 00 (2011) 000–000
δ i are
3
the relative change components in resistivity or resistance, σ i are stress components,
Δρij are the change components in resistivity, and ΔRij are the change components in resistance,
π 11 is
longitudinal piezoresistive coefficient,which indicates that the stress along a crystal axis will create effect of the relative changes components of in resistivity or resistance along the crystal axis, π 12 is transverse piezoresistive coefficient,which indicates that the stress along a crystal axis will create effect of the relative changes components in resistivity or resistance along the perpendicular crystal axis, π 44 is shear piezoresistive coefficient, which indicates that the shear stress will create effect of the relative changes components of in resistivity or resistance along the shear stress direction. The matrix of (3) is (4) {δ } = [π ]{σ } 3 Coordinate transformation expression of cubic crystal silicon piezoresistive coefficient matrix Four resistors (Fig.2 shows a piezoresistive pressure sensor) on the silicon diaphragm is of equivalent resistances R1 , R2 , R3 , R4 , which are located in four-side midpoints of the silicon diaphragm at maximum stress. The four resistors in circuit make up Wheatstone full bridge circuit and such changes of four resistors can be shown by voltage changes.
Fig.2.Piezoresistive pressure sensor[1].
In order to understand conveniently, we don’t transform the subscript temporarily. In Fig.3, the resistors R2 , R3 of sensor, current I and longitudinal stress σ L are along 1' axis; the resistors R1 , R4 of '
sensor and transverse σ T are along 2 axis. At the same time, the four resistors are located in O 1' 2 ' plane. We assume that the stress components σ 11 , σ 22 , σ 33 , σ 23 = σ 32 , σ 31 = σ 13 σ 12 = σ 21 of micro-unit at arbitrary point O are known. In O − 123 coordinates, the plane ABC connecting with coordinate plane by the point O is made up of tetrahedron OABC . We suppose that the outside normal of the plane ABC is along '
the axis 1 of the new coordinate system and the axis 2' , 3' are determined by the right-hand. Direction cosines are expressed as cos( 1' ,1)= l1 ,cos( 1' ,2)= m1 ,cos( 1' ,3)= n1 , cos( 2' ,1)= l2 ,cos( 2' ,2)=
m2 ,cos( 2' ,3)= n2 ,and cos( 3' ,1)= l3 ,cos( 3' ,2）= m3 ,cos( 3' ,3）= n3 .The matrix of direction cosines is an orthogonal matrix.
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Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
4
3
σ
2'
21
R1
,
11
σ
σ
p
12
22
1
,
R3
I
2
p σ
σο
R2
p
3
σ
L
R4
σ
1'
σ
T
13
2
σ
23
31
σ
32
1
3'
σ
33
Fig.3.Resistance position and stress of silicon diaphragm in the new coordinate system are expressed.
The stress relations are obtained by Cauchy stress relation and stress axis formula between old and new coordinates, which is expressed as ⎧ σ 11' ' ⎫ ⎡ l12 m12 n12 2m1n1 2l1n1 2l1m1 ⎤ ⎧σ 11 ⎫ ⎪σ ⎪ ⎢ 2 ⎥⎪ ⎪ 2 2 m2 n2 2m2 n2 2l2 n2 2l2 m2 ⎥ ⎪σ 22 ⎪ ⎪ 2 ' 2 ' ⎪ ⎢ l2 (5) ⎪⎪σ 3'3' ⎪⎪ ⎢ l32 m32 n32 2m3 n3 2l3 n3 2l3 m3 ⎥ ⎪⎪σ 33 ⎪⎪ . ⎥⎨ ⎬ ⎨ ⎬=⎢ ⎪σ 2'3' ⎪ ⎢l2l3 m2 m3 n2 n3 m2 n3 + m3 n2 l2 n3 + l3n2 l2 m3 + l3 m2 ⎥ ⎪σ 23 ⎪ ⎪σ 31' ' ⎪ ⎢ l1l3 m1m3 n1n3 m1n3 + m3 n1 l1n3 + l3n1 l1m3 + l3 m1 ⎥ ⎪σ 31 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎩⎪σ 1'2' ⎭⎪ ⎣⎢ l1l2 m1m2 n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ ⎪⎩σ 12 ⎪⎭ Formula (5) is simplified as
{σ } = [ M ]{σ } ,
(6)
'
where [ M ] is Bond stress transformation matrix[4,5]. According to the same principle, we can find the relationships of the relative change in resistivity between old and new coordinates ⎧ δ11 ⎫ ⎡ l12 m12 n12 2m1n1 2l1n1 2l1m1 ⎤ ⎧δ11 ⎫ ⎪δ ⎪ ⎢ 2 ⎥⎪ ⎪ 2 2 l m n 2 m n 2 l n 2 l2 m2 ⎥ ⎪δ 22 ⎪ 2 2 2 2 2 2 ⎪ 22 ⎪ ⎢ 2 2 (7) m32 n32 2m3n3 2l3 n3 2l3m3 ⎥ ⎪⎪δ 33 ⎪⎪ , ⎪⎪δ 3 3 ⎪⎪ ⎢ l3 ⎥⎨ ⎬ ⎨ ⎬=⎢ ⎪δ 2 3 ⎪ ⎢l2l3 m2 m3 n2 n3 m2 n3 + m3n2 l2 n3 + l3 n2 l2 m3 + l3m2 ⎥ ⎪δ 23 ⎪ ⎪ δ 31 ⎪ ⎢ l1l3 m1m3 n1n3 m1n3 + m3n1 l1n3 + l3 n1 l1m3 + l3 m1 ⎥ ⎪δ 31 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎩⎪δ12 ⎭⎪ ⎣⎢ l1l2 m1m2 n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ ⎪⎩δ12 ⎪⎭ Formula (7) can be expressed as ' '
' ' ' '
' ' ' '
' '
{δ } = [ M ]{δ }
(8)
'
Substituting (6), (8) to (4), {δ
{δ } = [ M ][π ][ M ] {σ } , '
−1
'
'
} is got (9)
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Guang YangH. and Hengyan Xie Engineering / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, Xie / Procedia 00 (2011) 000–000
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so the expression ⎡π ' ⎤ can be found ⎣ ⎦ −1 ⎡⎣π ' ⎤⎦ = [ M ][π ][ M ] .
(10)
Bond strain transformation matrix is ⎡ l12 m12 n12 m1n1 l1n1 l1m1 ⎤ ⎢ 2 ⎥ 2 2 m2 n2 m2 n2 l2 n2 l2 m2 ⎥ ⎢ l2 ⎢ l2 m32 n32 m3 n1 l3 n3 l3m3 ⎥ [N ] = ⎢ 3 ⎥ ⎢ 2l2l3 2m2 m3 2n2 n3 m2 n3 + m3n2 l2 n3 + l3n2 l2 m3 + l3m2 ⎥ ⎢ 2l l 2m m 2n n m n + m n l n + l n l m + l m ⎥ 1 3 1 3 1 3 3 1 1 3 3 1 1 3 3 1 ⎢ 13 ⎥ ⎢⎣ 2l1l2 2m1m2 2n1n2 m1n2 + m2 n1 l1n2 + l2 n1 l1m2 + l2 m1 ⎦⎥ By [ M ]−1 = [ N ]T ， ⎡π ' ⎤ is simplified
(11)
⎣ ⎦
⎡⎣π ' ⎤⎦ = [ M ][π ][ N ]
T
⎡ l12 ⎢ 2 ⎢ l2 ⎢ l2 =⎢ 3 ⎢l2l3 ⎢l l ⎢13 ⎢⎣ l1l2
m12
n12
2m1n1
2l1n1
m22 m32
n22 n32
2m2 n2 2m3 n3
2l2 n2 2l3n3
m2 m3 m1m3 m1m2
n2 n3 n1n3 n1n2
m2 n3 + m3 n2 m1n3 + m3 n1 m1n2 + m2 n1
l2 n3 + l3n2 l1n3 + l3n1 l1n2 + l2 n1
0 0 ⎤ ⎡ l12 ⎤ ⎡π 11 π 12 π 12 0 ⎢ ⎥⎢ 0 0 ⎥⎥ ⎢ l22 2l2 m2 ⎥ ⎢π 12 π 11 π 12 0 0 0 ⎥ ⎢ l32 2l3m3 ⎥ ⎢π 12 π 12 π 11 0 ⎥⎢ ⎥⎢ 0 0 π 44 0 0 ⎥ ⎢ 2l2l3 l2 m3 + l3m2 ⎥ ⎢ 0 ⎥ 0 0 0 π 44 0 ⎥ ⎢ 2l1l3 l1m3 + l3 m1 ⎢ 0 ⎥⎢ ⎥⎢ 0 0 0 0 π 44 ⎥⎦ ⎣⎢ 2l1l2 l1m2 + l2 m1 ⎦⎥ ⎢⎣ 0 2l1m1
m12
n12
m1n1
l1n1
m22 m32
n22 n32
m2 n2 m3 n1
l2 n2 l3n3
2m2 m3 2m1m3 2m1m2
2n2 n3 2n1n3 2n1n2
m2 n3 + m3n2 m1n3 + m3n1 m1n2 + m2 n1
l2 n3 + l3n2 l1n3 + l3n1 l1n2 + l2 n1
⎤ ⎥ l2 m2 ⎥ l3m3 ⎥ ⎥ l2 m3 + l3 m2 ⎥ l1m3 + l3m1 ⎥ ⎥ l1m2 + l2 m1 ⎦⎥ l1m1
T
(12)
4 The calculation of longitudinal piezoresistive coefficient π L and transverse piezoresistive coefficient π T Shown in Fig.2, the shape of silicon diaphragm on pressure sensor is a square. Under external pressure the bending deformation of the diaphragm is small and the thickness of resistance in comparision with the width and the length is negligible. So the force of the resistance can be approximated as a plane stress state σ 33 = σ 23 = σ 31 =0. Therefore, we only consider the longitudinal and transverse piezoresistive coefficient after coordinate transformation ( π L = π 11' , π T = π 12' ). Formula (14) using MATLAB matrix symbol evaluates is changed to transpi = [ (L1^2*p11+M1^2*p12+N1^2*p12)*conj(L1)^2+(L1^2*p12+M1^2*p11+N1^2*p12)*conj(M1)^2+(L1^2*p12+ M1^2*p12+N1^2*p11)*conj(N1)^2+2*M1*N1*p44*conj(M1*N1)+2*L1*N1*p44*conj(L1*N1)+2*M1*L1*p44*c onj(M1*L1), (L1^2*p11+M1^2*p12+N1^2*p12)*conj(L2)^2+(L1^2*p12+M1^2*p11+N1^2*p12)*conj(M2)^2+(L1^2*p12+M1^ 2*p12+N1^2*p11)*conj(N2)^2+2*M1*N1*p44*conj(M2*N2)+2*L1*N1*p44*conj(L2*N2)+2*M1*L1*p44*conj( M2*L2), …]
Here the author keeps the coordinate transformation calculations π 11 ， π 12 and omits other items. By '
the relationship l12 + m12 + n12 = 1 , the first item
'
π 11' of the calculation transpi is simplified to
2 2 2 2 2 2 π L = π 11' = π 11 − 2(π 11 − π 12 − π 44 )(l1 m1 + m1 n1 + l1 n1 )
By the relationship (l12 + m12 + n12 )(l22 + m22 + n22 ) = 1 and l1l2 + m1m2 + n1n2 = 0 , the second item the calculation transpi is simplified to π T = π 12' = π 12 + (π 11 − π 12 − π 44 )(l12l22 + m12 m22 + n12 n22 )
π 12' of
Guang Yang and Hengyan Xie / Procedia Engineering 29 (2012) 1612 – 1617 G. Yang, H. Xie/ Procedia Engineering 00 (2011) 000–000
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If we know the longitudinal stress component σ L and the transverse stress component σ T , the relative change of resistance can be got by the output of voltage through the Wheatstone full bridge circuit, by the item ΔR ≈ π σ + π σ . R
L
L
T
T
5Conclusions Piezoresistive coefficient matrix belonging to fourth-order tensor is similar to the stiffness matrix and co-mpliance matrix. If coordinate transformation matrix of piezoresistive coefficient with tensor in the old and the new coordinate system is very abstract, we derived the coordinate transformation of piezoresistive coefficient matrix by using an intuitive mechanical methods and simplified the transformation matrix with the Bond strain and stress transformation matrix. We obtained the formulas of the longitudinal and tra-nsverse piezoresistive coefficient π L , π T after symbolic Computation of MATLAB , which is useful to understand how the relative change in resistance are shown through output of the Wheatstone full bridge circuit. Acknowledgements This research is funded by the Research Fund for the Doctoral Program of Higher Education of China (Grant NO. 20102305120003). References [1] C. Liu,Foundations of MEMS,North-America,New Jersey,2006. [2] C. S. Smith,Piezoresistance effect in germanium and silicon,Phys Rev94(1954)42–49. [3] W.P.Mason and R.N.Thurston,Use of piezoresistive materials in the measurement of displacement, force, and torque,J. Acous. Soc. of Am. 29(1957)1096–1101. [4] W.Bond, The mathematics of the physical properties of crystals,BSTJ 22(1943)1-72. [5] B.A. Auld,Acoustic fields and waves in solids volume,1973.
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