The electrical engineering handbook CH051

The electrical engineering handbook

Amin, A “Piezoresistivity”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

Piezoresistivity

51.1 Introduction 51.2 Equation of State

51.4 Geometric Corrections and Elastoresistance Tensor 51.5 Multivalley Semiconductors

Directions 51.7 Semiconducting (PTCR) Perovskites 51.8 Thick Film Resistors

51.9 Design Considerations

51.1 Introduction

Piezoresistivity is a linear coupling between mechanical stress X kl and electrical resistivity rij Hence, it is represented by a fourth rank polar tensor Õijkl The piezoresistance properties of semiconducting silicon and germanium were discovered by Smith [1953] when he was verifying the form of their energy surfaces Piezore-sistance measurements can provide valuable insights concerning the conduction mechanisms in solids such as strain-induced carrier repopulation and intervalley scattering in multivalley semiconductors [Herring and Vogt, 1956], barrier tunneling in thick film resistors [Canali et al.,1980] and barrier raising in semiconducting positive temperature coefficient of resistivity (PTCR) perovskites [Amin, 1989] Piezoresistivity has also been investi-gated in compound semiconductors, thin metal films [Rajanna et al.,1990], polycrystalline silicon and germa-nium thin films [Onuma and Kamimura, 1988], heterogeneous solids [Carmona et al.,1987], and high T c

superconductors [Kennedy et al.,1989] Several sensors that utilize this phenomenon are commercially available

51.2 Equation of State

The equation of state of a crystal subjected to a stress X kland an electric field E i is conveniently formulated in the isothermal representation The difference between isothermal and adiabatic changes, however, is negligible [Keyes, 1960] Considering only infinitesimal deformations, where the linear theory of elasticity is valid, the electric field E i is expressed in terms of the current density I j and applied stress X kl as [Mason and Thurston, 1957]

Ei = Ei ( Ij, Xkl) i,j,k,l = 1,2,3 (51.1)

In what follows the summation convention over repeated indices in the same term is implied, and the letter subscripts assume the values 1, 2, and 3 unless stated otherwise Expanding in a McLaurin’s series about the origin (state of zero current and stress)

Ahmed Amin

Texas Instruments, Inc.

dEi = ( ¶ Ei/ ¶ Ij) dIj + ( ¶ Ei/ ¶ Xkl) dXkl

+ (1/2!) [( ¶2Ei/ ¶ Ij¶ Im) dIj dIm

+ ( ¶2Ei/ ¶ Xk l¶ Xno) dXkl dXno

+ 2 ( ¶2Ei/ ¶ Xk l¶ Ij) dXkl dIj] + H.O.T (51.2) The partial derivatives in the expansion Eq (51.2) have the following meanings: (¶E i/¶I j) = ri j (electric resistivity

tensor); (¶E i/¶X kl) = d ik l (converse piezoelectric tensor); (¶2E i/¶X k l¶I j) = (¶/¶X kl) (¶E i/¶I j) = Pijk l (piezoresistivity

tensor); (¶2E i/¶I j¶I m) = rijm (nonlinear resistivity tensor); (¶2E i/¶X k l¶X no) = diklno (nonlinear piezoelectric tensor)

Replacing the differentials in Eq (51.2) by the components themselves, we get

Ei = rijIj + diklXkl + 1/2 rijmIjIm + 1/2 diklnoXklXno + Õijkl Xkl Ij (51.3)

Most of the technologically important piezoresistive materials, e.g., silicon, germanium, and polycrystalline

films, are centrosymmetric The effect of center of symmetry (i.e., the inversion operator) on Eq (51.3) is to

force all odd rank tensor coefficients to zero; hence, the only contribution to the resistivity change under stress

will result from the piezoresistive term Therefore, Eq (51.3) takes the form

Ei = Sj rij Ij + SjSkSlPijkl Xkl Ij (51.4)

taking the partial derivatives of Eq (51.4) with respect to the current density I j and rearranging

¶Ei/¶Ij = rij(X) 2 rij(0) = SkSlPijkl Xkl

Thus, the specific change in resistivity with stress is given by

(drij/r0) = Pijkl Xkl (51.5) the piezoresistivity tensor Pijkl in Eq (51.5) has the dimensions of reciprocal stress (square meters per newton

in the MKS system of units) The effects of the intrinsic symmetry of the piezoresistivity tensor and the crystal

point group are discussed next

The transformation law of Pijkl (a fourth rank polar tensor) is as follows:

P¢ijkl = (¶x¢i/¶xm)(¶x¢j/¶xn)(¶x¢k/¶xo)(¶x¢l/¶xp)Pmnop (51.6) where the primed and unprimed components refer to the new and old coordinate systems, respectively, and

the determinants of the form \¶x¢ i/¶x m\, etc are the Jacobian of the transformation A general fourth rank

tensor has 81 independent components The piezoresistivity tensor Pijkl has the following internal symmetry:

Pijkl = Pijlk = Pjilk = Pjikl (51.7) which reduces the number of independent tensor components from 81 to 36 for the most general triclinic point

group C1(1) It is convenient to use the reduced (two subscript) matrix notation

where m,n = 1, 2, 3,…, 6 The relation between the subscripts in both notations is

Tensor: 11 22 33 23, 32 13, 31 12, 21 Matrix: 1 2 3 4 5 6 and

Pmn = 2Pijk l, for m and/or n = 4, 5, 6

Thus, for example, P1111 = P11, P1122 = P12, 2P2323 = P44, 2P1212 = P66, and 2P1112 = P16 Hence, Eq (51.5) takes the form

(dri/r0) = Pij Xj, (i, j = 1, 2,…,6) (51.9) Further reduction of the remaining 36 piezoresistivity tensor components is obtained by applying the generating elements of the point group to the piezoresistivity tensor transformation law Eq (51.6) and demanding invariance The following are two commonly encountered piezoresistivity matrices:

1 Cubic Oh (m3m): single crystal silicon and germanium

2 Spherical (¥ ¥ mmm): polycrystalline silicon and germanium and films

where P44 = 2(P11 2 P12) Thus, three coefficients P11, P12, and P44 are required to completely specify the piezoresistivity tensor for silicon and germanium single crystals, and only two, P11 and P12, for polycrystalline films Under hydrostatic pressure conditions, the piezoresistivity coefficient Ph for the preceding two symmetry groups is a linear combination of the longitudinal P11 and transverse P12 components, Ph = P11 + 2P12 Unlike

the elastic stiffness c ij (a fourth rank polar tensor), the piezoresistivity tensor Pmn is not symmetric, i.e., Pmn #

Pnm , except for the following point groups, C¥v(¥ ¥ mmm), Oh (m3m), T d (43m), and O(432).

51.4 Geometric Corrections and Elastoresistance Tensor

The experimentally derived quantity is the piezoresistance coefficient 1/R0(¶R/¶X) This must be corrected for the dimensional changes to obtain the piezoresistivity coefficient 1/r0(¶r/¶X) as follows:

1 Uniaxial tensile stress parallel to current flow

1/R0(¶R/¶X) –(s11 – 2s12) = 1/r0(¶r/¶x) = P11 (51.10)

P11 P12 P12 0 0 0

P11 P12 0 0 0

P44 0 0

P44 0 P¢44

P11 P12 P12 0 0 0

P11 P12 0 0 0

P44 0 0

P44 0

P44

2 Uniaxial tensile stress perpendicular to current flow

1/R0(¶R/¶X) + s11 = 1/r0(¶r/¶X) = P12 (51.11)

3 Hydrostatic pressure

1/R0(¶R/¶p) – (s11+ 2s12) = 1/r0(¶r/¶p) = Ph (51.12)

where s11 and s12 are the elastic compliances that appear in the linear elasticity equation x ij = s ijkl X kl , with x ij

the infinitesimal strain components Details on the different geometries and methods of measuring the piezore-sistance effect can be found in the References Equation (51.9) could be written in terms of the strain conjugate

x o as follows

(dri/r0) = Mio xo (51.13)

the dimensionless quantity M io is the elastoresistance tensor (known as the gage factor in the sensors literature)

It is related to the piezoresistivity Pik and the elastic stiffness c ko tensors by

thus, the 3 independent elastoresistance components (gauge factors) can be expressed as follows

M11 = P11 c11 + 2 P12 c12

M12 = P11 c12 + P12 (c11 + c12)

M44 = P44 c44

51.5 Multivalley Semiconductors

For a multivalley semiconductor, e.g., n-type silicon, the energy minima (ellipsoids of revolutions) of the

unstrained state in momentum space are along the six <100> cubic symmetry directions; they possess the

symmetry group O h (m3m) A tensile stress in the x-direction, for example, will strain the lattice in the xy-plane

and destroy the three-fold symmetry, thereby lifting the degeneracy of the energy minima However, the

four-fold symmetry along the x-direction will be preserved Thus, the two valleys along the direction of stress

will be shifted relative to the four valleys in the perpendicular directions

TABLE 51.1 Numerical Values of Pij and M ij for Selected Materials

Silicon

n-type 11.7 (W-cm) –102.2 53.4 –13.6 –72.6 86.4 –10.8

p-type 7.8 (W-cm) 6.6 –1.1 138.1 10.5 2.7 110

Ba.648Sr.35

Thin films

Thick film resistors

According to the deformation potential theory, the strain will shift the energy of all the states in a given band extremum by the same amount, i.e., the valley moves along the energy scale as a whole by an amount (the deformation potential constant) which is linearly proportional to the strain Let’s assume that the energy of

those on the y- and z-axes are lowered with respect to those on the x-axis This effect is represented by dashed

lines in Fig 51.1 As a result, there will be electron transfer from the high to low energy valleys The components

of the mobility tensor mxy (= e t/m xy , where e is the electron charge, t is the relaxation time, and m xy is the effective mass) are illustrated by arrows in Fig 51.1 The mobility anisotropy is due to the curvature of the

conduction band near the bottom The effective mass is inversely proportional to this curvature (1/m xy = (h/2p)–2 (¶2E/ ¶k x ¶k y ), which is larger for a direction perpendicular to the valley For an applied E field parallel to the

stress, the conductivity will increase (i.e., the resistivity decreases) relative to the unstressed state because of

the increase in the number of electrons in the four valleys (yz-plane) for which the mobility is large in the field

direction If the field is perpendicular to the stress, the conductivity will decrease (i.e., the resistivity increases) with stress Therefore, the piezoresistivity components P11 and P12 have opposite signs

A shear stress about the crystallographic axes will not lift the degeneracy; hence, P44 = 0 Similarly, a tensile stress along the <111> does not destroy the three-fold symmetry, and the degeneracy will not be lifted; thus,

no piezoresistance should be there Calculations based on the deformation potential model show that P11 =

22 P12, and P44 = 0

Information concerning the symmetry properties of the valleys can be derived from the representation surface

of the longitudinal P11 piezoresistance component This surface can be constructed by measuring the depen-dence of P11 on the crystallographic direction Smith showed that P11 is maximum in the <001> directions of

n-type silicon and not quite zero in the <111> directions Reasons for the deviation from the deformation

potential model of piezoresistivity in multivalley semiconductors are discussed in Keyes [1960] For n-type

germanium P11 is maximum in the <111> directions This is consistent with the loci of the valleys in these two materials Qualitatively, a piezoresistance effect is produced whenever the stress destroys the symmetry elements that are responsible for the degeneracy of the valleys

Intervalley scattering contribution to the piezoresistance of multivalley semiconductors may be comparable

to that of the strain-induced electron repopulation In this scattering process, the initial and final electron states

are in different valleys The effect of intervalley scattering can be deduced from the T–1 dependence of the elastoresistance tensor

FIGURE 51.1 Two-dimensional representation of the constant energy surfaces in momentum space of a multivalley

Piezoresistance effect in germanium and silicon, Phys Rev., vol 94, p 42, 1953 With permission.)

The influence of hydrostatic pressure on the electrical resistivity can provide additional insights on the transport properties Some of the noted features include (1) high pressures (in the GPa range, versus MPa for tensile stresses) can be applied without destroying the crystal; (2) it does not destroy the symmetry, provided

no phase transition is involved; hence, the symmetry degeneracies in the band structure are not lifted; (3) band edges which are not degenerate for symmetry reasons will be shifted; and (4) nonlinear effects could be discerned

Sensitivity Directions

Consider a long thin bar “strain gauge” cut from a piezoresistive crystal with the bar length parallel to an arbitrary direction in the crystal Let Pl, q, and j be the spherical coordinates of the longitudinal piezoresistivity tensor measured along the length of the bar For the cubic symmetry group Oh (m3m) of Si and Ge, Pl is given

by (Mason et al 1957)

Pl = P11 + 2(P44 + P12 - P11) [sin2q cos2q + cos4q cos2j sin2j].

= P11 + 2(P44 + P12 - P11) F[q,j]

The variation of Pl with direction may be considered as a property surface The distance from the center to any point in the surface is equal to the magnitude of Pl The function F[q,j] has a maximum for q = 54° 40`

and j = 45° which is the <111> family of directions, for which Pl takes the following form,

Pl = P11 + 2/3 (P44 + P12 - P11)

If (P44 +P12 – P11) and P11 have the same sign or 2/3 * (P44 +P12 – P11) * > P11 then the maximum sensitivity direction occurs along <111> If (P44 +P12 – P11) = 0 the longitudinal effect is isotropic and equal to P11 in all directions, otherwise it occurs along a crystal axis The maximum sensitivity directions are shown in Fig 51.2

for Si and Ge

51.7 Semiconducting (PTCR) Perovskites

Large hydrostatic piezoresistance Ph coefficients (two orders of magnitude larger than those of silicon and germanium) have been observed in this class of polycrystalline semiconductors [Sauer et al., 1959] PTCR compositions are synthesized by donor doping ferroelectric barium titanate BaTiO3, (Ba,Sr)TiO3, or (Ba,Pb)TiO3 with a trivalent element (e.g., yttrium) or a pentavalent element (e.g., niobium) Below the

ferroelectric transition temperature Tc, Schottky barriers between the conductive ceramic grains are neutralized

by the spontaneous polarization P s associated with the ferroelectric phase transition Above Tc the barrier height

increases rapidly with temperature (hence the electrical resistivity) because of the disappearance of P s and the decrease of the paraelectric state dielectric constant Analytic expressions that permit the computation of barrier heights under different elastic and thermal boundary conditions have been developed [Amin, 1989]

51.8 Thick Film Resistors

Thick film resistors consist of a conductive phase, e.g., rutile (RuO2), perovskite (BaRuO3), or pyrochlore (Pb2Ru2O7-x), and an insulating phase (e.g., lead borosilicate) dispersed in an organic vehicle They are formed

by screen printing on a substrate, usually alumina, followed by sintering at »850oC for 10 min

The increase of the piezoresistance properties of a commercial thick film resistor (ESL 2900 series) with sheet resistivity is illustrated in Fig 51.3 The experimentally observed properties such as the resistance increase and decrease with tensile and compressive strains, respectively, and the increase of the elastoresistance tensor with

sheet resistivity seem to support a barrier tunneling model [Canali et al., 1980].

FIGURE 51.2 Section of the longitudinal piezoresistivity surface, the maximum sensitivity directions in Si and Ge are shown [Keys, 1960].

FIGURE 51.3 Relative changes of resistance for compressive and tensile strain applied parallel to the current direction Note the increase of gage factor with sheet resistivity

1.0

0.8

0.6

0.4

0.2

0.2

1000 800

800 600

600 1000 400

400 200

200

0.4

0.6

0.8

1.0

1 K ½/

10 K ½/

100 K ½/

GFL = 5.5

GFL = 9.8

GFL = 13.8

R

R (%)

Tension

m Strains

m Strains

Compression

ESL 2900 Series

51.9 Design Considerations

Many commercially available sensors (pressure, acceleration, vibration,… etc.) are fabricated from piezoresistive materials (see for example, Chapter 56 in this handbook.) The most commonly used geometry for pressure sensors is the edge clamped diaphragm Four resistors are usually deposited on the diaphragm and connected

to form a Wheatstone bridge The deposition technique varies depending upon the piezoresistive material: standard IC technology and micro-machining for Si type diaphragms; sputtering for thin film metal strain gauges; bonding for wire strain gauges, and screen printing for thick film resistors Different types of diaphragms (sapphire, metallic, ceramic,… etc.) have been reported in the literature for hybrid sensors

To design a highly accurate and sensitive sensor, it is necessary to analyze the stress–strain response of the diaphragm using plate theory and finite element techniques to take into account: (1) elastic anisotropy of the diaphragm, (2) large deflections of plate (elastic non linearities), and (3) maximum sensitivity directions of the piezoresistivity coefficient Signal conditioning must be provided to compensate for temperature drifts of the gauge offset and sensitivity

Defining Terms

ri j: Electric resistivity tensor

d ik l: Converse piezoelectric tensor

Pijk l: Piezoresistivity tensor

rij m: Nonlinear resistivity tensor

dikln o: Nonlinear piezoelectric tensor

Related Topic

1.1 Resistors

References

A Amin, “Numerical computation of the piezoresistivity matrix elements for semiconducting perovskite

fer-roelectrics,” Phys Rev B, vol 40, 11603, 1989.

C Canali, D Malavasi, B Morten, M Prudenziati, and A.Taroni, “Piezoresistive effect in thick-film resistors,”

J Appl Phys., vol 51, 3282, 1980.

F Carmona, R Canet, and P Delhaes, “Piezoresistivity in heterogeneous solids,” J Appl Phys., vol 61, 2550,

1987

C Herring and E Vogt, “Transport and deformation-potential theory for many valley semiconductors with

anisotropic scattering,” Phys Rev., vol 101, 944, 1956.

R J Kennedy, W G Jenks, and L R Testardi, “Piezoresistance measurements of YBa2Cu3O7-x showing large

magnitude temporal anomalies between 100 and 300 K,” Phys Rev B, vol 40, 11313, 1989.

R W Keyes, “The effects of elastic deformation on the electrical conductivity of semiconductors,” Solid State

Phys., vol 11, 149, 1960.

W P Mason and R N Thurston, “Use of piezoresistive materials in the measurement of displacement, force,

and torque,” J Acoust Soc Am., vol 10, 1096, 1957.

Y Onuma and K K Kamimura, “Piezoresistive elements of polycrystalline semiconductor thin films,” Sensors

and Actuators, vol 13, 71, 1988.

K Rajanna, S Mohan, M M Nayak, and N Gunasekaran, “Thin film pressure transducer with manganese

film as the strain gauge,” Sensor and Actuators, vol A 24, 35, 1990.

H A Sauer, S S Flaschen, and D C Hoestery, “Piezoresistance and piezocapacitance effect in barium strontium

titanate ceramics,” J Am Ceram Soc., vol 42, 363, 1959.

C S Smith, “Piezoresistance effect in germanium and silicon,” Phys Rev., vol 94, 42, 1953.

Further Information

M Neuberger and S J Welles, Silicon, Electronic Properties Information Center, Hughes Aircraft Co., Culver

City, Calif., 1969 This reference contains a useful compilation of the piezoresistance properties of silicon

Electronic databases such as Chemical Abstracts will provide an update on the current research on

piezore-sistance materials and properties