piezoresistance effect of silicon

8/2/2019 Piezoresistance Effect of Silicon

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Sensors and Actuators A, 28 (1991) 83-91 83

PiezoresistanceYozo Kanda

effect of silicon

Hamamatsu University School of Medicine, Hamamatsu 431-31 (Japan)(Received August 28, 199D; n revised form December 17, 1990; accepted January 17, 1991)

AbstractThe principle of the piezoresistance. effect (PR) of n- and p_Si is explained by the carrier-transfermechanism and the effective mass change. The origin of the shear piezoresistance coefficient Q inn-Si is also a stress-induced effective mass change. Agraphical representation of the PR on crystallographicorientations and the effect of impurity concentration on the PR are given for n- and p-Si. The non-linearity of the PR is also mentioned.

1. IntroductionRecently, the piezoresistance effect in sil-icon [l] has been re-examined from the fol-lowing viewpoints:(1) The application of this effect to variousmechanical sensors has been widely extended.The reasons for this are its high sensitivityand good linearity, the superior mechanicalproperties of silicon, the ease of mass pro-duction by micromachining and the ease ofintegration in standard IC technology (whichoffers the possibility of on-chip signal con-ditioning). With the requirement by sensorengineers for higher precision sensors hascome the need for a precise knowledge ofthe piezoresistance effect.(2) As MOS devices are scaled down tosubmicron dimensions, new reliability prob-lems arise. One such problem is a new physicalphenomenon considered to be related to thefact that mechanical stress is more significantfor submicron device/process design. This ismainly due to several kinds of device deg-radation resulting from process-induced me-chanical stress, which cause additional reli-ability problems. This trend calls for a deeperphysical understanding and re-examination ofVLSI reliability from the aspect of mechanicalstress.This paper focuses on the former viewpoint,and discusses the work accomplished to dateand newly obtained rtsults. The following

sections give a brief explanation of the prin-ciple, phenomenological description, and nec-essary data for design, including the effectof impurity concentration and non-linearity.

2. PrincipleThe resistance R0 of a rectangular con-ductor is expressed by

where po is the resistivity and 1, w and t arethe length, width and thickness of the con-ductor, respectively. When the resistor isstretched, the relative change in resistanceis given byAR Al Aw At---- +PR,=7 w t poIntroducing Poissons ratio A, whereAw At Al-=-=-A-

W t 1

the gauge factor G (strain sensitivity) is

(2)

G_ W&I- - =l+ZA+ iE!@lE E (3)

where E= AZ/I s the strain. The first two termsin eqn. (3) represent the change in resistancedue to dimensional changes (dominant for


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metal gauges), while the last term representsthe change in resistivity (dominant for semi-conductor gauges). In semiconductor gauges,the resistivity change is larger than the di-mensional change by about a factor of 50,and the latter is therefore neglected hereafter.If a uniaxial stress T is applied along thedirection of current flow, the piezoresistanceratio is given by

where T, is the longitudinal piezoresistancecoefficient and T the longitudinal stress. Therelation between rrr and G isG=Y,T, (5)where Y, s Youngs modulus and is anisotropicwith respect to crystal direction [2]_Many-valley energy surfaces for n-type sil-icon in k (wave vector) space are shown inFigs. 1 and 2 [3]. Wave vector k is relatedto the momentum P by P =hk /2?r, where his Plan&s constant. Silicon has three pairsof valleys. These surfaces consist of ellipsoidsof revolution located on the cube axes. Theeffective masses of electrons in a single valleyare anisotropic, m ,, > m I, and hence the mo-bilities in the valley are also anisotropicCL~ k. L=cLJk =4,rJm.r,, =m,,/m,,assuming the relaxation times 7, =rll.

co101

Fig. 1. Schematic diagram of the (100) and (010) valleysin k-space for a-Z& Dotted lines show effect of stress. (a)corresponds to stress TI and (b) to Te

c0101Q TI


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electrons no does not change. The relativechange of the population of the valleys onthe ith axes caused by applying an arbitrarilyoriented uniaxial stress T is given by [3]A?F/n = - [(q+ l/3]

where (Ye= 86.8&-5.0)/m and m is the freeelectron mass. Therefore, the shear piezo-resistance component IQ, is given by [7]- aom II844

7T44= 1+2L (14)

x (~k3@)(~11--4T (8)where L? s the shear deformation potential,kB is Boltzmanns constant, 0 is the tem-perature in K, si, are the compliance constants,qti) is a unit vector pointing from the centreof the Brillouin zone to the ith valley and tis a unit vector in the direction of the tension.The longitudinal piezoresistance componentrll is given by [4, 51

22 C&l d(l -L)--=- 3k,O 1+2L (9)

Energy surfaces for p-type silicon in k spaceare very complex, and different from thosefor n-Si, as shown in Fig. 3(a). The bandedge, the upper P,, state, consists of a pairof two-fold degenerate bands at k=O usuallydesignated as the light and heavy holebands. These energy surfaces are warpedspheres. The spin-orbit split-off band, thelower Pin state, has a spherical energy surface.When a uniaxial tensile stress is applied par-allel to the (111) direction, the degeneracyWhen the electric field is applied perpen-dicular to the direction of the, applied stress,the conductivity is expressed by(T= #)pL +n()~,, +nC3)jq,)e (10)and then the transverse piezoresistance com-ponent q2 is easily given by712= - 7T11/2 (11)

Next, we consider the shear piezoresistance

k

component q.,. When a shear stress, T,, is Yapplied in the plane normal to the (OOl), heband edges for the (100) and (010) valleysremain unchanged as shown by arrangement(b) in Fig. 1, but the effective mass of the Cal(001) valley changes as shown by arrangement(b) in Fig. 2 due to the special character of Ed?the conduction band edge of Si [6]. Whenan electric field is applied to a four-terminaldevice parallel to (lOO), a transverse voltagewhich is proportional to T6 is generated be-tween two output electrodes due to the asym-metry of the (001) effective mass to the (100) _direction. The proportionality constant de-fines ~~1VA= ndT6 or p61~=mT6 (12)Under a shear strain e, the electron ellipsoid _ _ _ __perpendicular to the shear plane will be dis- =1torted and the band energy becomes [6] J_~s E1/2- tt =LI

@ISTiESS =

I+ aoh2e,kxk,, (13) Fig. 3. Schematic diagram of the energy surfaces in k-space for p-Si (a) without stress and (b) under stress.


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86of the valence band is lifted and two bandsof prolate and oblate ellipsoidal energy sur-faces with anisotropic mass parameters areformed as shown in Fig. 3(b) [8, 91. Con-sequently, the resistivity change comes fromboth the mass changes and hole transfer. Theband splitting E, and the effective massesare given by [8]d= ZD:s,T3 (15)mlm ,, =x+Y~,

m h , = x - 2~~m/mzl = y1 - y3(1 - 4e/A)m/mzI1= y1 ++a(1 - 4EM)

(164

(16b)Even if the effective masses change, the den-sity of state effective masses m13Rremainunchanged [8], and m: is approximatelyequal to mZZ3 lo]. Assuming that all holesof the acceptor levels are exhausted with andwithout stress, the total number of conductionholesp, does not change. It is also assumedthat the scattering time 7;: s isotropic andrl = Q. The conductivity (J of p-type Si along(111) is(r= ((l) t +pQ) & )e2Therefore, the longitudinal piezoresistancecoefficient is given by [lo]

2YsD:hT1= 3y, (17)where A is the spin-orbit splitting energy,0: is the deformation potential constant andthe yis are the effective mass parameters. Forthe electric Geld perpendicular to (ill), thetransverse piezoresistance coefficient is alsogiven byrr,= - rJ2 (18)For a more accurate description [ll] nu-merical analysis is needed and the contriibution of the inter-valley scattering effect tothe piezoresistance effect [4], which was ne-glected for brevity, should be taken into ac-count in the above-mentioned mechanismsfor both n- and p-type Si.

3. Phenomenological descriptionThe first-order piezoresistance is expressed

bYAPiitR= XVjktTti (19)k,1where 7~~ is the component of the piezo-resistance tensor and T,-- s the componentof the stress tensor. In the commonly usedsix-component notation, noting that subscripts11, 22, 33, 13, 12, correspond to 1, 2, 3, 4,5, 6, respectively, eqn. (19) is rewritten as

and the tensor is given by(20)

(21)(a) Dependence on crystallographic a.&When we refer to the crystallographic axesin the symmetry of silicon, (m3m), the fun-damental piezoresistance coefficients are rll(longitudinal), 7r12 (transverse) and 7r44(shear). Consider a general case in Cartesiancoordinates of arbitrary orientation withprimed quantities. Assuming a plane stress,namely neglecting the effect of stress com-ponent T;, the resistivity components pi understress are expressed by

(224(22t-J)

&=p,,(&T; + a&T;+ r&T:) (224Here, Ti and T; are the normal stressesparallel and perpendicular to the current,respectively, and Tk is the in-plane shearstress. Among rljs in eqn. (22), three typicalpiezoresistance effects will be consideredwhen stress is applied in the material. Thefirst is the longitudinal piezoresistance coef-ficient, when the current and field are in the


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direction of the uniaxial stress, denoted byl-I,; another is a transverse piezoresistancecoefficient, when the current and field areperpendicular to the stress, denoted by l-I,;and the third is a shear piezoresistance coef-ficient II,. These three coefficients are givenby [12, 131

IT;* = I-l, = 7r l2 + (n* - w12 - 74 4)

x (11*~22+m1*m*2+n12n*2) (239

(23s)The graphs of room-temperature l-I,, II,and IIJ2, as a function of crystal directionfor orientations in the (001) plane, are shownin Figs. 4-7 [5, 141.The upper halves of thegraphs represent positive values of the piezo-resistance coefficient (i.e., the resistivity in-creases with tensile stress) and the lowerhalves, negative values of the piezoresistancecoefficient (i.e., the resistivity decreases with

tensile stress). In the graphs for l-I,, theorientation means the direction of the current.Graphs are in units of lo- Pa- based onthe data of Smith [l] (shown in Table 1).

4

Fig. 4. Room-temperature piezoresistance coefficients inthe (001) plane of n-Si (lo- Pa-).

Fig. 5. Room-temperature piezoresistance coefficients inthe (001) plane of p_Si (lo- Pa-).

Fig. 6. Piezoresistance coefficient &J2 in the (001) planeof n-Si (lo- Pa-).

Fig. 7. Piezoresistance coefficient a&n in the (001) planeof p-Si (lo- Pa-).

(b) Imp wi ly concent ra t ion e f f ec tT h e piezoresistance coefficient lJ(N, S)with an impurity concentration N and at atemperature 0 can be rewritten in the form

l-I(N, 0) =P(N, @l-I (300 K) (24)where P(N, 0) is the piezoresistance factorgiven by


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88TABLE 1. Deformation potential constant (eV), piezo-resistance components (lo- Pa-), elastic complianceconstant (lo- Pa-) and mass parameters

n-type p-typeP (a cm)

n,, (1O-n Pa-)qz (lo- Pa-)nM (lo-* Pa-)2, (eV)3 (ev)D. (eV)D: (eV)D: (ev)A (cV)mllimm,im7 1Yz7 3q1 (10-r Pa-)srz (lo- Pa-)sM (lo- Pa-)

11.7 7.8- 102.2 +6.6+.53.4 - 1.1- 13.6 + 138.1-5.28.5

3.44.43.90.0440.91610.1905 4.260.381.560.768- 0.2141.26

Fig. 8. Piezoresistance factor P(N, S) as a function ofimpurity concentration and temperature for n-Si.

The Fermi integral is a function of temper-ature and Fermi energy assuming that therelaxation time 7 is a function of energy E,T= TV,!? [5]. q., in n-Si is excluded from theabove discussion. Of course, the resistivityalso depends on the impurity concentrationand temperature. Graphs of P(N, 0) areshown in Figs. 8 and 9. Sensors have usuallybeen made by diffision since Tufte and Stel-zers pioneering work [14]. Their results areshown in Figs. 10-13. The average piezo-

Fig. 9. Piezoresistance factor P(N, ) s a function ofimpurity concentration and temperature for psi.

Fig. 10. Variation of nr, in n-type Si-diffised layers withtemperature and surface concentration. Concentrations arein WI-~.resistance coefficient of a diffused layer iswritten as [14]

where x is the distance from the surface ofthe layer, x0 the layer thickness, and V(X) ndu(x) the piezoresistance coefficient and con-ductivity due to the impurity distribution fora depth X. ?Y s only slightly larger than thebulk r having an impurity concentration equalto the surface concentration of the diffusedlayer (see Fig. 14 [15]).Conventional pressure sensors are made byp-type diffused layers. What is an optimum


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L -A_J.-1 * I I1 -a0 -60 -10 .20 0 10 .O 60 80 mo

Fig. 11. Variation of n@ in p-type Si-diised layers withtemperature and surface concentration (in cm-). Thea~~p~n 7rM=39qt2has been made to obtain thesevalues.

I

.r** -80 -60 -40 -20 0 20 40 60 80 100

Fig. 12 Variation of the diffused-layer resistance withtemperature for n-type Si layers at different surface con-centrations (in crnq3}.

Fig. 13. Variation of the dised-layer resistance withtemperature for p-type Si layers at different surface con-centrations (in UII-~),

SURfkcf Colitis (krous~cuFig. 14. Buik piezoresistive coefficient u& vs. impurity~n~ntration in p-type silicon with (111) orientation anddiffusedcccfficients ?r{,vs. surface impurity ~~centratjonfor Gaussian and erfc profiles of p-type impurities.

OS 40\. SObr 20Xzi 10a 0

-10-20

lOI8 1o* 1020 102Nr(cm-J)

Fig. 15. TCR(u) and TCG@) vs. surface concentration inp-type diEused layers.

concentration for p-type sensors? The resis-tance change AR under strain Q isAR=R,(l+rr AO)G,(1+/3 A+

~~~G~~l~(a~~~ A@})E (27)where ar is the temperature sufficient ofresistance (TCR) and p is the temperaturecoefficient of the gauge factor (TCG).From eqn. (27) it can be seen that first-order temperat~e effects are removed ifWI-p==0 (28)

cu,p and of p for p-type silicon are plottedas a function of surface concentration, N, inFig. 15 [16, 171,The conditions which satisfyeqn. (28) are obtained around N,= 10 andlO**cme3. For n-type silicon, on the otherhand, no such condition could be found.


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(c) Non-linear effectIn general, the piezoresistance effect of Sihas better linearity fir transducing a me-chanical signal into an electrical one thanother effects, However, for high-precision sen-sors a more detailed knowledge of the non-linearity (NL) is required. It is convenient,from an application point of view, to define

COMPRESSION . TENSION(MPd (MPd

4-OdC0:wc

-10- A:&(a) LONGITUDINAL MODE

IO3a

-10 013OCA lOC

@I TRANSVERSE MODEFig. 16. Temperature dependence of the non-linearity ofthe piezoresistance effect of p_Si in (110) stress. Impuritysurface concentration is 2~10~ cm-. (a) Longitudinalmode, (b) transverse mode.

the NL by [18]NLV, T,)

WnJ -WV(29)in which R(T) denotes the resistance at stressT, and T,,, the maximum applied stress. NLalso represents the difference from the lin-earity divided by the resistance change atmaximum stress. Since most pressure sensorshave p-type gauges along [llO], the NL ofp-Si for (110) stress is most important, andis thus shown in Fig. 16 [19]. Let us considerthe PR coefficient up to second order:

where rWAVre the second-order PR com-ponents. The symmetry of the diamond struc-ture reduces the number of independent IT@~values to nine. The first- and second-orderpiezoresistance coefficients in some typicalconfigurations are given in Table 2. Exper-imentally obtained second-order piezoresis-tance tensor components of p- and n-type Siare given in Table 3 [20, 211.

4. DiscussionIt has long been believed that the piezo-resistance effect of n-Si was completely ex-plained by the electron transfer mechanism[4]. However, an important puzzle has re-mained unsolved: on the basis of the many-valley model, the Q coefficient is shown to

TABLE 2. The first- and second-order piezoresistance coefficients in some tyPiCal configurationsDirection Configuration PR coefficientsStress Current Fist order Second order(100) (100) L Tll WI 1(100) (010) T =I2 %3(110) (l!O) L (ru+ ?r,z + WY2 ~Ir,11+~1~+~~112+71166+41r661~~4(110) (110) T (mu+ r12 - mw (ml1 + mz+ 2n12 f m66 -4%d4(110) (0001) T 7712 m 22(111) (111) L (~+2~,2+2~)/3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(111) (ii2) T (77,,272,~-Q4)/3 (Ql+ 2W1~2+j~+2nlu+ W+2~~-2rr*c,4i k, , -2T,,6)/9


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91TABLE 3. Summary of the experimental second-order PRtensor components of n- and p_Si in unit of lo-* MPa-*.Carrier concentration at surface, 1 X 10 cm-. A, assumedzero: N, not measuredSecond-order PRtensor component Conductivity type

P nAA-22A98A-5144-51

71-36-5-35ANNNN

be zero by the electron transfer mechanism,but this is different from experimental results[l, 201.The discrepancy can be explained bythe effective mass change of the valley per-pendicular to the shear plane. The crosssection of the effective mass changes from acircle to an ellipse. The effective mass changecould be measured only by cyclotron reso-nance at liquid helium temperature. Now, wecan see the effective mass change throughthe piezoresistance measurement at roomtemperature. From eqn. (14) n-U s indepen-dent of temperature. This is a very novelfinding, In Section 3, the author dared toshow the piezoresistance data of n-S, whichare rarely used, because of the possibility ofmaking attractive complementary gauges sat-isfying eqn. (28). The non-linear physics ofthe piezoresistance effect has just begun. Ifthe NL is an odd function of stress as shownin Fig. 16(a), the third-order term is dominant,which has not been studied yet. If the NLis an even function of stress as shown in Fig.16(b), the second-order term plays a dominantrole.

References1 C. S. Smith, Piezoresistance effect in germanium andsilicon, Phys. Rev., 94 (1954) 42-49.

2 J. J. Wortman and R. A. Evans, Youngs modulus,shear modulus, and Poissons ratio in silicon and ger-manium, /. Appl. Phys., 36 (1965) 153-156.

3 Y. Kanda, Effect of stress on germanium and siliconp-n junctions, Jpn. J. Appl. Phys., 6 (1967) 475-486.4 C. Herring and E. Vogt, Transport and deformation-potential theory for many valley semiconductors withanisotropic scattering, Phys. Rev., 102 (1956) 944-961.5 Y. Kanda, A graphical representation of the piezo-

resistance coefficients in siliwn, IEEE Tmns. ElectronDev ices, ED-29 (1982) 64-70.

6 J. C. Hensel, H. Hasegawa and M. Nakayana, Cyclotronresonance in uniaxially stressed silicon. II. Nature ofthe covalent bond, Phy s. Rev ., 138 (1965) A 2256238.

7 Y. Kanda and K. Suzuki, Origin of the shear piezo-resistance coefficient of n-type silicon, Phys. Rev. B,43 (1991) 67.54-6756.8 H. Hasegawa, Theory of cyclotron resonance in strainedsilicon crystals, Phys. Rev., 229 (1963) 1029-1040.9 J. C. Hensel and G. Feher, Cyclotron resonance ex-periments in uniaxially stressed silicon: valence bandinverse mass parameters and deformation potentials,Phys. Rev., 129 (1963) 1041-1062.

10 K. Suzuki, H. Hasegawa and Y. Kanda, Origin of thelinear and nonlinear piezoresistance effects in p-typesilicon, Jpn. J. Appl . Phys., 23 (1984) L 871-L874.11 J. T. Ienkkeri, Nonlinear effects in the piezoresistivityof p-type silicon, Phys. Status Solidi (b), 136 (1986)373-385.

12 W. G. Pfann and R. N. Thurston, Semiconductor stresstransducers utilizing the transverse and shear piezo-resistance effect, J. Appl. Phys., 32 (1961) 2008-2019.13 Y. Kanda, Graphical representation of the piezore-sistance coefficient in silicon-shear coefficient in plane,Jpn. J. Appl. Phys., 26 (1987) 1031-1033.14 0. N. Tufte and E. L. Stelzer. Piezoresistive orooertiesof silicon diffused layers, i Appl. Phjx, 34 -(1963)313-318.

15 D. R. Kerr and A. G. Milnes, Piezoresistance of diffusedlayers in cubic semiconductor, J. Appl. Phys., 34 (1963)727-731.16 Anthony D. Kurtz, ISA 22 P4-I-PAID-1967.17 M. Shimazoe, K. Yamada and Y. Takehashi, Tem-perature characteristics of semiconductor strain gauges,Ext.Abstr., 36th Autumn Meet., Jpn. Sot. Appl. Phys.,Fukuokq Japan, 1975, 62, 24aD6.18 K. Yamada, M. Nishihara, S. Shimada, M. Tanabe,M. Shimazoe and Y. Matsuoka, Nonlinearity of thepiezoresistance effect of p-type silicon diffused layers,IEEE Trans . Electron Devices, ED-29 (1982) 71-77.

19 K. Matsuda, Y. Kanda, K. Yamamura and K. Suzuki,Nonlinearity of piezoresistance effect in p- and n-typesilicon, Sensors and Actuat ors, AZ -A23 (1990) 45-48.20 K. Matsuda, Y. Kanda and K. Suzuki, Second-orderpiezoresistance coefficients of n-type silicon, Jpn. J.

Appl. Phys., 28 (1989) L1676-L1677.21 K. Matsuda, Y. Kanda, K. Yamamura and K. Suzuki,Second-order piezoresistance of p-Si, Jpn. J. App[. Phy s.,29 (1990) L1941-L1942.

piezoresistance effect of silicon

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