  Purpose: Developing photon-counting CT detectors requires understanding the
impact of parameters such as converter material, absorption length and pixel
size. We apply a novel linear-systems framework, incorporating spatial and
energy resolution, to study realistic silicon (Si) and cadmium telluride (CdTe)
detectors at low count rate. Approach: We compared CdTe detector designs with
$0.5\times0.5\; \mathrm{mm}^2$ and $0.225\times0.225\; \mathrm{mm}^2$ pixels
and Si detector designs with $0.5\times0.5\; \mathrm{mm}^2$ pixels of 30 and 60
mm active absorption length, with and without tungsten scatter blockers.
Monte-Carlo simulations of photon transport were used together with Gaussian
charge sharing models fitted to published data. Results: For detection in a 300
mm thick object at 120 kVp, the 0.5 mm and 0.225 mm pixel CdTe systems have
28-41 $\%$ and 5-29 $\%$ higher DQE, respectively, than the 60 mm Si system
with tungsten, whereas the corresponding numbers for two-material decomposition
are 2 $\%$ lower to 11 $\%$ higher DQE and 31-54 $\%$ lower DQE compared to Si.
We also show that combining these detectors with dual-spectrum acquisition is
beneficial. Conclusions: In the low-count-rate regime, CdTe detector systems
outperform the Si systems for detection tasks, while silicon outperforms one or
both of the CdTe systems for material decomposition.
