Experimental quantitative study into the effects of electromigration field moderation on step bunching instability development on Si(111)

We experimentally studied the effects of a moderated electromigration field on the dynamics of the step bunching process on the Si(111) surface at 1130 °C (regime II) and 1270 °C (regime III). The surfaces with step bunch morphologies were created by annealing vicinal Si(111) at fixed temperatures while the applied electric field $E$ was adjusted for every experiment. Scaling relations, $y_m~h^{\alpha }{E}^q$, between the slope of a step bunch $y_m$, step bunch height $h$, and electromigration field $E$ were experimentally probed. Scaling exponents $\alpha$ ≈ 2/3 and $q$ ≈ 1/3 were extracted from the step bunch morphologies created by annealing Si(111) in the regime III (1270 °C), which are in good agreement with the predictions of the generalized BCF theory. Scaling exponents $\alpha$ ≈ 3/5 and $q$ ≈ 1/3 were extracted from the morphologies created by annealing in regime II (1130 °C). This result was compared to the scaling relations derived within the frame of the transparent step model, which correctly predicts the formation of the step bunching instability by step-up adatom electromigration. The scaling relation obtained by experiment was found to differ from the model predictions. We measured values of critical electric field ($E_{\mathrm{cr}}$), i.e., minimum electric field required for the step bunching to take place. A relatively weak field of $E$ > 0.5 V/cm was found to be sufficient to initiate the step bunching process in regime II. This contrasts with regime III, where $E_{\mathrm{cr}}=1.0$ and 2.0 V/cm were measured for Si miscut from the (111) plane by 1.1° and 2.5°, respectively. The increased values of $E_{\mathrm{cr}}$ were attributed to the enhanced step-step repulsion in regime III. The theoretically predicted formation of compressed step density waves was observed upon annealing in both regimes with $E<E_{\mathrm{cr}}$.

Figure 1

Schematic of a sample holder inside the crucible. The effects of in-plane applied electric field and sample temperature on step bunching were separated in this way.

Figure 2

Step bunching morphologies created on Si(111) by annealing at 1130 °C with different applied electric fields. The surface is off cut 2.5° towards the [11-2] direction. The direction of a miscut is from left to right in all images as shown by a stairway sign in Fig. 2(a). Darker areas correspond to step bunches. (a) Differentiated AFM image of a step bunched Si(111) obtained by dc annealing; $E$$=3.9\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, annealing current $I$ $=$ 2.4 A. (b) Differentiated AFM image of Si(111) after annealing with $E$ $=$ $2.4\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 1.5 A. (c) Phase AFM image of Si(111) after annealing with $E$ $=$ $1\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 0.6 A. There is an obvious widening of step bunches as compared to annealing exclusively by dc current. (d) Phase AFM image obtained after annealing with $E$ $=$ $0.9\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 0.55 A. Most of the step bunches expand to the size of 3–3.8 μm and cover approximately 70% of the surface area. (e) Phase AFM image obtained after annealing with $E$ $=$ $0.9\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 0.4 A. The step bunches in some areas are still separated by terraces. In other areas they “consume” the terraces and reach the neighboring bunches. (f) Differentiated AFM image of Si(111) after annealing with $E$ $=$ $0.5\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 0.3 A. The surface appears to be stable against the step bunching when the applied electric field is lower than $0.5\hspace{0.5em}\mathrm{V}/\mathrm{cm}$.

Figure 3

Selected curves of maximum slope $y_m$ versus the step bunch height ($h$) for vicinal Si(111) annealed at 1130 °C with different electromigration fields $E$ (every fourth experimental point is shown for clarity). The $y_m$($h$) data ($E$ $=$ const) were fit to a power-law function $y_m=y_0{h}^{\alpha }$ and values of $\alpha$ ranging between 0.57 and 0.59 were found with a fitting error of ±0.03.

Figure 4

Cross-sectional and slope profiles along the miscut direction of 208–210-nm-high step bunches produced by annealing with different applied electric fields $E$. The step bunches created by annealing with stronger electromigration fields have steeper slopes.

Figure 5

Maximum slope $y_m$ as a function of electromigration field $E$ for 250- and 130-nm-high step bunches. Fitting data to a power-law function $y_m=y_1{E}^q$ results in the value of $q$ ≈ 1/3. Dashed lines show the curve fits for data calculated from the experimentally obtained scaling relations of $y_m=y_0{h}^{\alpha }$ ($E$ $=$ const), while the solid lines are the curve fits for step bunch slopes taken directly from the AFM measurements.

Figure 6

Step bunching morphologies created on Si(111) at 1270 °C by annealing with different electromigration fields. The surface is off cut 1.1° towards the [11-2] direction. The direction of the miscut is from left to right in all images. Darker regions correspond to step bunches. (a) Differentiated AFM image of a step bunched Si(111) surface obtained by dc annealing with $E$ $=3.6\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, annealing current $I$ $=$ 3.8 A. (b) Differentiated AFM image of Si(111) after annealing with $E$ $=2.4\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 2.1 A. (c) Phase AFM image obtained after annealing with $E$ $=1.3\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 1.35 A. (d) Differentiated AFM image obtained after annealing at $E$ $=$ 1.2 V/cm, $I$ $=$ 1.3 A. (e) Phase AFM image obtained after annealing at $E$ $=$ $1.1\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 1.2 A, which shows a transitional surface morphology. Most step bunches expand as far as the neighboring bunches, while flat 0.7–0.8-μm-wide terraces are still present on the surface. (f) Differentiated AFM image obtained after annealing with $E$ $=$ $0.8\hspace{0.5em}\mathrm{V}/\mathrm{cm}$, $I$ $=$ 0.85 A. The surface appears to be stable against the step bunching.

Figure 7

Cross-sectional and slope profiles along the miscut direction of 60-nm-high step bunches produced by annealing at 1270 °C with different applied electric fields $E$. The step bunches created by annealing with weaker electromigration fields are significantly wider and have gentler slopes.

Figure 8

Selected curves of maximum slope $y_m$ versus the step bunch height $h$ for Si(111) surface annealed at 1270 °C with different electromigration fields $E$. The $y_m$($h$) data were fit to a power-law function $y_m=y_0{h}^{\alpha }$ and values of $\alpha$ ranging between 0.64 and 0.67 with a maximum fitting error ±0.03 were found (only curves for selected values of $E$ are presented).

Figure 9

Maximum slope $y_m$ as a function of electromigration field $E$ for 65- and 40-nm-high step bunches. Fitting the data to a power-law function $y_m=y_1{E}^q$ results in a value of $q$ ≈ 1/3. Dashed lines show the curve fits for data calculated from the experimentally obtained scaling equations of $y_m=y_0{h}^{\alpha }$ ($E$ $=$ const); solid lines are the curve fits for step bunch slopes taken directly from the AFM measurements.

Figure 10

Selected curves of step bunch width $L$ versus the step bunch height $h$ for Si(111) surface annealed at 1270 °C with different electromigration fields $E$. A power-law fit to the data yields $L~h^{0.42\pm 0.03}$.