Understanding Smart Sensors - Nomads.usp

52 Understanding Smart Sensorsalternative design demonstrates one of the different possibilities that semiconductortechnology brings to sensing.3.2.2 Piezoresistivity in SiliconThe analytic description of the piezoresistive effect in cubic silicon can bereduced to two equations that demonstrate the first-order effects [3].∆E = P ⋅ I( p X + p X )(3.3)1 0 1 11 1 12 2∆E = P ⋅I⋅p X(3.4)2 0 2 44 6where ∆E 1 and ∆E 2 are electric field flux density, P 0 is the unstressed bulk resistivityof silicon, I is the excitation current density, π is the piezoresistive coefficient,X 1 and X 2 are axial stress tensors, and X 6 is a shear stress tensor due to theapplied force.The effect described by (3.3) is utilized in a silicon pressure transducer ofthe Wheatstone-bridge type. Regardless of whether the sensor designer choosesn-type or p-type layers for the diffused sensing element, the piezoresistive coefficientsp 11 and p 12 in (3.3) will have opposite signs. That implies that, throughcareful placement, orientation with respect to the proper crystallographic axis,and a sufficiently large aspect ratio for the resistors themselves, it is possible tofabricate resistors on the same diaphragm that both increase and decrease,respectively, from their nominal values with the application of stress.The effect described by (3.4) is typically neglected as parasitic in thedesign of a Wheatstone-bridge device. A closer look at (3.4) reveals that theincremental electrical field flux density, ∆E 2 , due to the applied stress, X 6 ,ismonotonically increasing for increasing X 6 . In fact, (3.4) predicts an extremelylinear output, since it depends on only one piezoresistive coefficient and oneapplied stress. Furthermore, the incremental electric field can be measured by asingle stress-sensitive element. That is the theoretical basis for the design of thetransverse voltage or shear stress piezoresistive strain gauge.Figure 3.2 shows the construction of a device that optimizes the piezoresistiveeffect of (3.4). The diaphragm is anisotropically etched from a siliconsubstrate. The piezoresistive element is a single, four-terminal strain gaugelocated at the midpoint of the edge of the square diaphragm at an angle of45 degrees. The orientation of 45 degrees and the location at the center of theedge of the diaphragm maximizes the sensitivity to shear stress (X 6 ) and theshear stress being sensed by the transducer by maximizing the piezoresistivecoefficient, p 44 .