Page 166 - Organic Electronics in Sensors and Biotechnology

Integrated Pyr oelectric Sensors 143

The second important equation is the spatial heat current density
J which for the nth layer is defined as

∂ Tx()
Jx() =−δ n (4.28)
n n x ∂

The problem can be solved analytically with respective boundary
conditions at each layer, basically yielding the average temperature
Tt() for the nth layer.
n
According to Fig. 4.14, the pyroelectric layer is the second layer
and is given by

dQ dQ dT
ω
I pyro = = ⋅ = p pyro A T (4.29)
dt dT dt
with T being the average temperature of the pyroelectric layer that is
represented by a complicated matrix equation 32, 33 accounting for the
material constants (δ, c, d) and for the heat input which is described
by the absorption coefficient η, referring to the amount of light
absorbed by the sensor and the heat radiation transfer coefficient g ,
H
taking into account the heat loss per unit area at the front and back
sides of the sensor.
By means of the equivalent circuit, the voltage response can be
calculated from the differential equation

∂ T ∂ V V
p pyro A = C + (4.30)
t ∂ ∂ T R

with the solution

ω
V t () = pA R T ⋅ = I ⋅ R (4.31)
pyro + ω 2 2 2 pyro + ω 2 2 2
1 RC 1 RC
This solution has already been given in the beginning of Sec. Phe-
nomenology of the Pyroelectric Response and describes the absolute
value of the pyroelectric voltage response that can be measured by a
lock-in amplifier.

Comparison with Experiment Before using the model to test the influ-
ence of certain material parameters, a comparison between the
experimental and the modeled results for one example is given
(with the material properties specified in Fig. 4.15). The shape of the
curve can be explained very well by the one-dimensional multilayer
model over four orders of magnitude in frequency. Some of the