Next, run the sweep "temperature" for a fixed value of the TX gate voltage (e.g. 3.3V) and a fixed value of sense voltage ( also 3.3 V). The dark current can now be plotted as a function of temperature for a chosen bias on sense. Figure 5 shows the Arrhenius plot of the current density, which can be used to estimate the contribution from the various physical processes. The generation current from the space charge layer is proportional to ni (the intrinsic density), which can be expressed as
$$ n_{i}=\sqrt{N_{C} N_{V}} e^{\frac{E_{G}}{2 K T}} $$
Assuming that the effective density of states is independent of temperature (this is not true):
$$ \begin{array}{l}{\frac{d}{d\left(\frac{1000}{T}\right)} \log _{10} n_{i} \approx\frac{\log _{10} e E_{G}}{2000 k_{B}}\left(\frac{1000}{T}\right)} \\ {\rightarrow E_{G}=\frac{2000 k_{B}}{\log _{10} e} \frac{\Delta I}{\Delta s}}\end{array} $$
where ΔI/Δs is the slope of the logscale plot. In the simulation below, Eg ~ 1.36eV, which of course is larger than the band gap of silicon (1.12eV), indicating that diffusion also contributes to the dark current.
Run the following lines of script in the script prompt to plot the dark current for temperature:
# Physical constants:
#
qe = 1.602176487e19;
kB = 1.3806504e23;
Ccg = 1e15; # 1fF
texp = 1e3; # 1ms
#
# Get sweep data
#
T = pinch(getsweepdata("temperature","T"));
Is = pinch(getsweepdata("temperature","Isub"))/qe;
#
# Calculate and plot results
#
vdark = qe/Ccg*sqrt(Is*texp);
plot(T,Is/1000,"T (K)","Dark current (ke/s)");
plot(T,vdark*1e3,"T (K)","Dark current shot noise (mV)");
invT = 1000/T;
logIs = log10(Is);
plot(invT,logIs,"1000/T (1/K)","Dark current (e/s), log scale","Dark Current Arrhenuis plot");
sizet = size(T);
nt = sizet(1);
delta_ni = (logIs(nt)  logIs(1))/(invT(nt)  invT(1));
Eg = 2000*kB*delta_ni/log10(exp(1))/qe;
?"Eg = " + num2str(Eg) + " eV";
The Arrhenius plot of the dark current density at Vsense=Vtx= 3.3V is shown to the right. The slope of the curve gives a bandgap of 1.36eV, which indicates contributions from both the generation current in the space charge layer (proportional to ni) and from the diffusion current (proportional to ni2)
Alternatively, one can set the sense voltage to a range of DC values and plot slices of the sweep results for each sense voltage.
