Quantification of Secondary Electron Doping Contrast in the Scanning Electron Microscope on 4H-SiC

We discuss the quantification of the secondary electron doping contrast in the scanningelectron microscope on 4H-SiC. It has been observed and studied at length mostly on silicon, but noconclusive theoretical model has been proposed yet. Therefore, we propose a simple physical modelthat allows for a quantification of the doping contrast. It is based on the changes in effective ionizationenergy for different doping concentrations and types. For a better agreement between our model andthe experiment, a locally increased temperature of the electron system or separate quasi Fermi levelsfor electrons and holes have to be assumed. A line profile of the sample under investigation is comparedwith a SRIM simulation of the corresponding implant and shows very good agreement.


Introduction
To enable the development of future improved SiC power devices, the knowledge of the effective doping concentration and electric field distribution is crucial. Due to the sub-micrometer scales of interest, a very good spatial resolution is required. Our method of choice is the secondary electron (SE) doping contrast in the scanning electron microscope (SEM) which shows clear advantages compared to atomic force microscopy (AFM) based approaches, such as scanning capacitance microscopy (SCM) and scanning spreading resistance microscopy (SSRM), such as the increased spatial resolution, simpler sample preparation and scalability for production use.
The SE doping contrast is usually understood as the effect, where acceptor doped regions appear brighter than donor doped areas in the SEM. It has already been described by other authors in many publications including, but not limited to [1][2][3][4][5][6][7][8][9][10][11][12][13]. Most of these publications have focused on silicon and described the doping contrast qualitatively. Ref. [5] has investigated the doping contrast on 4H-SiC and [10] has attempted a quantification using a Monte-Carlo simulation and achieved a qualitatively good agreement with the experiment. The most prevalent mechanisms proposed for the SE doping contrast are surface band bending [8], differences in work function [9] and stray electric fields at the sample surface [5]. Many of the previous papers have focused on maximizing the SE doping contrast in an effort to identify and better understand the mechanisms contributing to it. Here, we develop a method that allows for the 8 or 16 bit gray values of SEM images to be converted to a doping concentration via a simple physical model.

Model and Conversion
Our approach relies on the differences in effective ionization energy. The model is based on the assumption that the primary electron beam generates electron-hole pairs in the sample. The generated electron-hole pairs fill the surface states and suppress the surface band bending. This leads to flat-band condition, which is similar to the flat-band condition caused by ultraviolet light radiation [14]. The model is similar to the one discussed in [2], but with key differences being the aforementioned flatband condition, the addition of a higher local temperature of the electron system and quasi Fermi level splitting. In addition, a small positive sample bias is introduced to shift the SE spectrum for monotonic contrast. The primary origin of the SEs, that reach the detector of the microscope, is the valence band. This can be safely assumed, as even at very high n-doping concentrations of 10 20 cm −3 the number of electrons in the conduction band is magnitudes smaller than the concentration of atoms (~5 × 10 22 cm −3 ) with multiple valence electrons each.
The SEs that are ionized from the sample still need to pass the detection energy barrier E detect to get detected. This position of E detect above the vacuum level E vac can be varied by adding a positive bias to the sample stage in the SEM. Our model is summarized in Fig. 1. The electrons originating from the valence band need to have enough energy to not only get ejected from the sample by passing E vac , but also pass E detect . A smaller ∆E effective will result in a larger number of SEs that can be detected. Because its exact value is not known precisely, we need to express the ∆E effective differences between n-and p-doped areas through a known quantity to quantify the doping contrast: The potential difference between E detect and the Fermi level E f is constant across the whole sample. Therefore, we subtract it, reducing ∆E effective to ∆E v , which is defined as the energetic difference between E f and ∆E v : To convert the gray values of the SEM image, we need to calculate ∆E v , which can be calculated from E f using Eq. (2). E f is calculated by solving the charge neutrality equation

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Silicon Carbide and Advanced Materials To convert the gray values to doping concentrations, two regions in the SEM image, where the doping concentration is known, have to be marked by the user. Their gray values are then linked to their corresponding ∆E v . Now each gray value can be linearly inter-or extrapolated to obtain the corresponding ∆E v . By inverting the relation in Fig. 2 the doping concentration can be obtained. These two steps can be summarized in a conversion function as shown in Fig. 3.
Due to the generation of electron-hole pairs by the primary electrons of the electron beam, the conversion assuming room temperature grossly over-and underestimates the aluminum doping concentration for input gray values above and below the reference, respectively. Therefore we introduce two approaches to take the additional electron-hole pairs and their effects on the calculation of E f into account. In the first approach, we assume a higher local temperature of the electron system. For the measurement presented later in this paper and many others a temperature of 800 K fits the reference well. Alternatively, the generation of electron-hole pairs can be viewed as local quasi Fermi level splitting ∆E f , similar to the effect found in photonic devices [15]. The quasi Fermi level splitting is defined symmetrically. The levels for electrons and holes are shifted by ∆E f /2 from the reference Fermi level. For the same measurement, the best agreeing ∆E f was found to be 1.6 eV. This is consistent across different SiC samples when using the same primary electron beam current and calibrations can be easily reused from previous measurements. Increasing the beam current results in a larger quasi Fermi level splitting and a higher temperature needed for this fit. Introducing quasi Fermi level splitting leads to a modification of Eq. (3): ∆E v can be recalculated with these corrections and is shown in Fig. 2. This results in the conversion functions of Fig. 3. These two functions have a similar gradient around the Al reference doping concentration of 10 19 cm −3 . The discrepancies can be explained by the fundamental differences between the two mod-els describing the additional electron-hole pairs. The incompatibility and limited validity of some phenomenological models involved in the calculation of the functions may also contribute. The dashed lines correspond to the two gray value and concentration pairs that were marked by the user. The vertical section at a gray value of 125 is caused by ∆E v being identical for very low n-and p-doping concentrations.

Materials and Methods
For the acquisition of the below shown images a Hitachi SU8230 SEM was used. The sample was cleaved in air and attached to the sample holder with silver glue. The silver glue was dried in a nitrogen flooded oven at 60°C. For an exact description of the sample see [16]. An acceleration voltage of 1 kV with an extraction current of 5 µA was used. A positive stage bias of 2.5 V was applied to the sample with an external voltage supply. This is necessary to suppress artifacts at the p-n junction in the SEM image. In order to minimize the effects of carbon depositions on the sample surface, each area for imaging was only scanned once when taking the image.

Results and Discussion
In Fig. 4 the algorithm for the conversion from gray values to doping concentration is applied to a line profile of the sample as shown in Fig. 5 (a). It is compared with a SRIM simulation (Stopping Range of Ions in Matter) [17] and both the approaches with a higher temperature and quasi Fermi level splitting fit the implant with a peak at 0.7 µm well. At around 0.1 µm the 10 20 cm −3 concentration peak plateau better agrees with quasi Fermi level splitting. The differences between the concentration profiles of the SEM image and the SRIM concentration profile can be explained by inaccuracies of the SRIM simulation when compared to the ion implantation and subsequent annealing. The deviation at 1 µm is explained by the beginning of the p-n junction, where ∆E v is mainly influenced by the space charge region. In Fig. 5 (b) the same algorithm is applied to the SEM image of Fig. 5 (a). The values between concentrations of 10 15 cm −3 aluminum and 10 15 cm −3 nitrogen are plotted white for a better visibility of doping concentrations in the relevant scales.

Silicon Carbide and Advanced Materials
For both Figs. 4 and 5, the aluminum concentration values below 10 17 cm −3 near the space charge region of the p-n junction must be interpreted carefully, as they are likely to be already influenced by the junction.  Fig. 5 (a). The conversion used the functions for a temperature of 800 K and for a ∆E f of 1.6 eV of Fig. 3 and was averaged over a width of 400 pixels. The profiles are compared to the doping concentration profile given by a SRIM simulation.  Fig. 5: (a) shows a section of a SEM image of a sample from [16] and (b) shows the same section converted to a doping concentration map with a quasi Fermi level splitting of 1.6 eV. The reference points for the conversion are also indicated and correspond to the dashed lines of Fig. 3. The positive values of the colorbar correspond to the log 10 of the p-doping (Al) concentration and the negative values correspond to the n-doping (N) concentration.

Conclusion
We developed a simple physical model to convert the gray values of a SEM image to doping concentrations. The SEM image taken with a positive sample bias of 2.5 V was converted to a map of the doping concentration and the vertical line profile is in good agreement with a SRIM simulation.
Two approaches to account for the imbalance caused by extra electron hole pairs induced by the primary electron beam were developed. The model using quasi Fermi level splitting is in slightly better agreement with the data than the model assuming a higher temperature of the electron system.
Exploiting a variation of the imaging parameters and with a better understanding of the physical processes involved, a quantification of the secondary electron doping contrast without prior knowledge of the sample will be possible.