Optimal detectability combined with picometre range precision to position light atoms from
HR STEM
images
Authors: Gonnissen Julie , De Backer Annick , Jan den Dekker Arnold , Sijbers Jan , Van Aert Sandra 
Published: December 2016

In the past few years a lot of research has been done to improve the imaging power to detect light atoms like oxygen, lithium, and hydrogen, since they play a key‐role in interesting industrial applications such as lithium‐batteries or hydrogen‐storage materials. Since material properties crucially depend on the
exact
atomic arrangement, an estimation of the atomic column positions with picometre range precision is needed, which is feasible using HR STEM [1]. It is investigated if a single optimal design can be found to both detect and position light atoms. The principles of statistical detection theory [2] are used to quantify the so‐called probability of error P
e
, in a binary hypothesis test. P
e
can be computed using realistic simulations to describe the experimental images [3] and can be used to optimise the experimental settings for the detection of light atoms from HR STEM images, as shown in [4]. To determine the optimal experiment design to position light atoms, use is made of the concept of Fisher information. The attainable precision with which unknown continuous structure parameters can be estimated is given by the lower bound on the variance with which an unknown parameter can be estimated from a set of observations, which is given by the so‐called Cramér‐Rao Lower Bound (CRLB) [5]. The optimal statistical experiment design of a HR STEM experiment for positioning light atoms is given by the microscope settings that minimise this CRLB. For both research questions, it will not only be investigated where in the detector plane the most sensitive region is located, but moreover, precise optimal inner and outer STEM detector angles can be derived quantitatively. The ultimate goal is then not to achieve optimal visual interpretability, but to obtain quantitatively the optimal experiment design for which the unknown structure parameters are obtained with the highest possible precision.

To illustrate the concept, the problem of suggesting optimal detector settings to detect and position the oxygen atoms in SrTiO
3
is considered, as well as detecting and positioning the lithium atoms in LiV
2
O
4
. A 4.66nm thick LiV
2
O
4
crystal is therefore simulated for an incoming electron dose of 10
5
e
‐
/Å², and a 1.95nm thick SrTiO
3
crystal is simulated, using an incoming electron dose of 10
4
e
‐
/Å². For the detection problem, a binary hypothesis test is performed where both hypotheses correspond to either the presence or absence of the oxygen or lithium atoms in the crystal. P
e
is computed as a function of the STEM inner and outer detector angles of which results are shown in Fig.1(a) for SrTiO
3
and in Fig.1(b) for LiV
2
O
4
. For the positioning problem, the CRLB is computed as a function of the STEM inner and outer detector angles of which results are shown in Fig.2(a) for SrTiO
3
and in Fig.2(b) for LiV
2
O
4
. The same optimal detector angles for the detection and positioning problem are found, which lie in the low angle ADF STEM regime for both applications, for a probe semi‐convergence angle of 21mrad. To detect and position oxygen in SrTiO
3
, the optimal detector range is 21–100mrad, while for the detection and positioning of Li in LiV
2
O
4
the optimal detector settings are 23–26mrad.

In conclusion, it is demonstrated that the experiment design can be optimised in order to detect and position light elements with the highest possible precision. Consistent optimal designs are found for both problems. It can be shown that picometre range precision is feasible for the estimation of the atom positions using an appropriate incoming electron dose at the optimal experimental settings to detect the light atoms.