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SrTiO₃ (001) etching

The etching speed of step edges was simulated the same way as the etch pit development. The starting images had step edge angles of 0, 15, 30, and 45 degrees, as shown in Fig. 1. The starting image size was 128 x 128 pixels. Etching probability was only changed for straight sites p₁. All other sites had their etching probabilities set to 1. This effectively changed the straight site to kink site etching probability ratio.

Thus step morphologies of surfaces obtained after a Monte Carlo simulations are shown in Fig. 2.

After simulation, some of the images were used for further analysis. The step edge positions were detected and the distance of each edge from the original position was calculated. A typical image with detected step edges is shown in Fig. 3.

The following table shows the distance how much each step edges moved. The distances were obtained by comparing the initial and final images obtained from the simulation. The positions are scaled by simulation steps (time).

Step edge positions

angle	p(1)	step edge distance							average
0	0.1	21.9429	26.9814	25.6484	25.4748	26.7322	25.0039	25.9773	25.3944
0	0.01	8.0625	8.2884	7.6334	8.6151	8.2042	8.2440	8.3362	8.1977
0	0.001	2.6250	2.0593	2.2370	2.5571	2.3281	2.3776	2.1670	2.3359
0	0.0001	0.6088	0.6492	0.6287	0.4577	0.5851	0.5857	0.4941	0.5728
15	0.1		26.1660	27.8479	27.3875	26.6967	26.2981	26.6698	26.8443
15	0.01	14.2935	14.9279	14.9732	15.0272	14.2381	15.7863	14.1450	14.7702
15	0.001	11.7392	13.7615	12.8702	13.6050	13.4825	15.3411	12.9248	13.3892
15	0.0001	11.9239	13.6187	13.7880	13.5587	13.7710	15.0201	12.6994	13.4828
30	0.1	32.2489	28.6247	29.4410	27.9324	29.1386	29.2814	28.8890	29.3651
30	0.01	20.9512	20.5174	20.5011	20.9657	20.9164	22.0435	21.2306	21.0180
30	0.001	21.9377	22.4043	19.5154	20.4571	20.5773	20.3992	21.1453	20.9195
30	0.0001	20.4384	20.9713	19.5677	20.7928	20.3855	20.9407	21.0193	20.5880
45	0.1	26.8701	32.6394	29.5323	29.6501	30.1106	30.7656	28.9832	29.7930
45	0.05	25.4558	27.1731	26.6674	25.9429	26.0109	25.4734	25.8082	26.0760
45	0.01	24.3952	23.8884	23.8324	21.3961	23.0715	22.9010	22.9643	23.2070
45	0.001	21.9203	24.3176	23.2743	22.1668	22.9344	23.8601	24.1047	23.2255
45	0.0001	23.1774	24.7460	23.8568	22.4493	22.6969	22.9944	24.3146	23.4622

It is interesting to compare these reuslts with the etch pit etching speeds. The simulation image size was different, but the pit etching results can be scaled to give comparable edge movement speeds:

p(1)	pit edge speed	scaled pit edge speed
0.1	6.1735	24.6939
0.05	4.4034	17.6136
0.01	2.0217	8.0868
0.001	0.5363	2.1452
0.0001	0.1421	0.5684

What we would expect is that the step edge etching speed is (sin(a)+cos(a)) times faster than the etch pit edge etching speed, where a is the step edge angle. The etch pit is, of course, aligned with the crystallographic axes (the 0 degree direction). The following table shows the ratios of simulated step edge etching speeds and pit etching speeds for various angles, together with the straight edge site etching probabilities p₁.

Angle	sin(a)+cos(a)	p(1)=0.1	p(1)=0.05	p(1)=0.01	p(1)=0.001	p(1)=0.0001
0	1.000	1.028		1.014	1.089	1.008
15	1.225	1.087		1.826	6.241	23.721
30	1.366	1.189		2.599	9.752	36.221
45	1.414	1.206	1.480	2.870	10.827	41.278

What this table shows is that for a probability ratio of p₁/p₂=0.1 the simulated step edge movement speed is lower than the scaled etch pit edge movement speed. For this p₁/p₂ ratio, the step etching rate is too low. The difference is most clearly visible for the 45 degree step orientation.

On the other hand, the step edge etching is too fast for a p₁/p₂ ratio of 0.01. A suitable value appears to be somewhere in between, at around a ratio of 0.05. In other words, it appears that to obtain consistent step edge and pit edge etching ratios, the etching of a straight edge site should be about 16 times less likely than the etching of a kink site.

This value (16) also shows why the in situ etching AFM observation only shows linear etch pit growth. This happens because for a relatively small p₁/p₂ ratio the quadratic growth region ends very quickly, before the etch pit shapes becomes observable by AFM (on the order of 20 to 50 unit cells).

An interesting question is the influence of pH on the etching probabilities. A change in pH can, presumably, change the p₁/p₂ etching ratio and therefore cause the surface to become more wavy or more faceted. This is indeed observed in actual etched surfaces.