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An application of the kinematic theory to copper crystal etching studies
J. Phys. Cha.
Solids
Pergamon
Press 1965. Vol. 26, pp. 1287-1292.
AN APPLICATION
Printed in Great Britain.
OF THE KINEMATIC
COPPER CRYSTAL ETCHING L. D. HULETT, Solid State Division,
Jr. and
F. W. YOUNG,
Oak Ridge National Laboratory, (Receiued
1 February
THEORY
TO
STUDIES* Jr.
Oak Ridge, Tennessee
1965)
Abstract-The kinematic theory of crystal dissolution predicts the shape of a dissolving crystal as a function of time, and suggests a convenient technique for measuring reaction rate as a function of crystallographic orientation. It has been applied to a study of dislocation etch pit growth on (111) surfaces of copper undergoing anodic dissolution in solutions containing HCI and HBr. The dissolution shapes of these pits at successive times have been determined from interference photomicrographs taken as the reaction progressed. The evolutions of the shapes have been shown to be in accord with predictions of the theory, and reaction rates as functions of orientations near (111) have been determined for various conditions of dissolution.
1. INTRODUCTION
THE surface of a crystal may be described by closepacked terraces separated by steps of atomic height, and dissolution of such a surface occurs by the removal of atoms from the steps, resulting in the progression of these steps across the surface. The so-called kinematic theory of crystal dissolution (or growth) which was proposed by FRANK(~) and CABEERA@) describes the dissolution in terms of the flow of these steps. While CHERNOV(3) has shown that this theory can be applied also to non close-packed surfaces, we will be concerned with close-packed and vicinal surfaces and so will follow the earlier treatments. This kinematic theory of crystal dissolution has been used by CABRERA@) to describe the formation of etch pits at dislocation sites during dissolution. When the step flux J at a given point on the surface is a function only of the step density p at that point, the kinematic theory predicts that points of a given orientation (p) will have straight line trajectories as dissolution proceeds. If the observation of such straight line trajectories for a crystal undergoing dissolution indicates that the kinematic theory applies for that system, this elegant theory can * Research sponsored Commission Corporation.
under
by the U.S. Atomic Energy contract with the Union Carbide
then be used in the description and elucidation of the dissolution. Frank and Cabrera define step density as proportional to the slope of a surface (with respect to a close-packed plane), then treat the surface as if it were continuous. While MULLINS and HIRTH@) have recently reformulated the theory, taking into account the discontinuous nature of stepped surfaces, the present paper is concerned only with the continuum theory. Dissolution studies are greatly simplified for systems to which the kinematic theory is applicable. IVES@) first demonstrated the quantitative applicability of the theory in the dissolution of LiF, and IVES and FRANK@) also used it to treat BATTERMAN’S work(7) on the dissolution of germanium. The authors have studied the growth of dislocation etch pits on (111) surfaces of copper undergoing anodic dissolution in solutions containing HCl and HBr.@) In the present paper we make a more rigorous application of the kinematic theory of dissolution to this process in order to interpret the evolution of the topography of the pits. 2. EXPERIMENTAL
PROCEDURE
The specimens, prepared by the methods of YOUNG and SAVAGE,@) were highly-perfect (ZOO1287
1288
L.
D. HULETT,
Jr.
1000 dislocations/ems) single crystal wafers, 2.5 cm dia. x O-7 cm thick, of 99.999% pure copper. They were mounted in silicone rubber bases so that only the (111) orientation on the flat side of the wafer was exposed to the solution. Etching was done anodically in mixtures of HCl and HBr solutions. The experimental set-up, similar to that used by the authors in previous work,@) was such that the crystal surface could be observed with an optical microscope during etching. The growth of the etch pits was recorded by time-lapse photomicrography using an interference microscope and a 3.5 mm automatic camera. The concentration of HCl was always 6 M, the concentration of HBr was varied in the range 0.03 M to 1.0 M, and the current densities ranged from S-30 mA/cms.
3. RESULTS Figures l(a) and l(b) illustrate the two general types of dislocation pits that were observed. The pit shown in Fig. l(a), having steeper sides, was generated by a dislocation that nucleated steps faster than the dislocation that generated the pit seen in Fig. l(b). They are called deep and shallow pits, respectively. The deep pitswere used in studying the motion of vicinal surfaces 3-8” from (ill), while the shallower pits were used to study surfaces of 14-3” misorientation. No data could be taken on surfaces below IQ” misorientation. The montage in Fig. 2 illustrates the growth of a deep pit. Neither of the two types of pits had completely smooth sides, for there was always a certain amount of ledge structure. This structure, which was usually more apparent on the lower misoriented varied with dissolution conditions; surfaces, generally the average ledge height was N 300 A. However, for the purpose of this paper, the existence of ledges was ignored, and measured step densities represent average misorientations of the surface. Ledges will be discussed further in a later publication. Pit profiles in the (211) directions were constructed from interference photomicrographs as in Fig. 3(a). A family of curves of this type could be obtained from photomicrographs taken at successive times as the pit grew in size as dissolution proceeded. Differentiating the curve in Fig. 3(a) produced the p vs. x curve in Fig. 3(b), which
and F. W. YOUNG,
Jr.
represents the distribution of step density along the side of the pit. From a time-dependent family of curves of the type shown in Fig. 3(b), x vs. time trajectories could be determined for fixed values of P*
Figure 4 is a plot of the trajectories of various step densities measured from the two types of pits shown in Fig. 1 for a current density of 10 mA/cms in 6 M HCl-0.25 M HBr solution. The linearity of these trajectories indicate that the kinematic theory of dissolution is applicable, since Cabrera and Frank show that a necessary condition for the applicability of this theory is that x-time trajectories be linear. From the theory, the slope of the trajectory of a given step density pl is equal to (aJ/8p)jpl. Since J is a function of p only, the slope of the trajectory must be constant. From the slopes of the trajectories the J vs. p curve can be calculated : P
J
=
sSdp+J(po)
(1)
PO
where
=
slope of the trajectory
of a point of density p.
Figure 5(a) is a plot of the slopes of the trajectories shown in Fig. 4. By graphical integration of the curve in Fig. 5(a), the J vs. p curve in Fig. 5(b) was obtained. The integration constant, J(po), was determined from the motion of the discontinuity formed at the intersection of the bottom of a flat-bottomed pit with the pit side. Figure 6 is a photograph of such a flat-bottomed pit. The dislocation that caused this pit was moved from this site so that nucleation stopped, and as the pit sides continued to be attacked they left a wake of almost perfectly oriented (111) surface. The velocity of the discontinuity in the x-direction and the step density of the pit side that made up the discontinuity were measured simultaneously. From these measurements the step flux JO corresponding to the step density po of the pit side at the point of intersection was calculated. From the kinematic theory of
FIG. 1. Interference photomicrographs of deep pit (a) and shallow pit (b) generated at dislocation sites on (111) surface of copper. Dissolution conditions: 6 M HCl, 0.25 M HBr, 10 m,4/cm2, 50-set dissolution time. Magnification = 990 x .
500
steps/p
5 i*o
Y
1”
T ‘0 P
(a)
/
x
[ii21
FIG. 3. Construction
of the profile (a) of a dislocation pit and the derivation distribution (b).
of the step density
FIG.
0.
Interferrnce
photomicrograph of fat-bottomed 900 x Magnitication
pit.
FIG, 8. Evolution
of a treilinp-edge discontinuity Dissolution conditions:
in a pit growing under the conditions of a Type I step flux vs. step 6 M HCI, 0.03 I.%1HBr, 5 m.~/em2. .50+x frame intervals.
density
curve.
2
APPLICATION
OF THE
KINEMATIC
THEORY
TO
COPPER
CRYSTAL
310
100
ETCHING
270
STUDIES
1289
210
510 t-----
FIG. 4. X-time
dissolution by:
trajectories along which step density is constant. Step micron) associated with each trajectory is indicated.
the velocity of a discontinuity
V PO.Pl
$0) =
is given
- Jh)
PO-P1
(2)
where po and pl are the step densities of the two surfaces that make up the discontinuity, and J(p0) and J(pl) are the respective step fluxes. In the case of the discontinuity at the bottom of a flat-bottomed pit such as in Fig. 6, the J and p of the (111) surface are both zero. Then, J(Po) Vpo,o = PO From the experimentally determined values of Vp0.0 and po the value of J(po), the integration constant, was calculated. It has been found that the x-time trajectories along which p is constant are straight lines for current densities 5-20 mA/cms in solutions with bromide ion concentrations of 0.03-1.0 M. For a current density of 5 mA/cms, J vs. p curves for
density
(steps/
bromide ion concentrations of 0.03, O-25, and 1 *OM were determined, see Fig. 7. For small concentrations of bromide ion (@J/ap2) is negative for all p (curve A, Fig. 7), while for higher bromide ion concentration the curves have inflection points such that (SJ/ap2) is positive for small values of p (curves B and C). Following CABRERA@) we denote curve A as Type I, and curves B and C as Type II. The series of photographs and their corresponding p-x curves shown in Fig. 8 demonstrate the evolution of a trailing-edge discontinuity. An early variation in nucleation rate caused the profile of this pit to be sigmoid. After the second frame the nucleation rate had completely stopped, and the pit was flat-bottomed. The shallow slopes originally generated behind overtook and compromised the larger slopes in the mid-portion of the profile. This pit was grown under conditions corresponding to a Type I curve (curve A in Fig. 7). Note that a region of shallow slope was generated at the periphery of this pit. We have obtained some information on the effect of changing the current density, in a solution
L.
1290
D. HULETT,
Jr.
of given bromide concentration, on the J-p curves; compare the curve in Fig. 5(b), 10 mA~cm2, with curve 3 in Fig. 7,s mA/cms. But, for current densities higher than 10 mA/cms there were difficulties in determining the J-p curves. The too,
I
/
I
I
I
1
and
F.
W.
YOUNG,
10 mA/cms the higher, better defined slope was regenerated. When the pit was in the ‘flatbottomed’ state the dislocation that caused it had not stopped nucleating, as in the case of the pit in Fig. 8. In fact, the nucleation rate was higher at 30 mA/cma than at 10 mA/cms. Note also that a region of very shallow slope, which was propagated much faster than the remainder of the pit, was generated at the periphery when the current density was 30 mA/cma. When the current density was returned to 10 mA/cms, the shallow periphery stopped moving and eventually disappeared. The ‘flat-bottomed’ state of the pit was rather extraordinary. Only a smali percentage of the dislocation pits behaved this way. The slopes at the bottoms of most pits were increased by an increase in current density. The generation of the shallow periphery was observed for all pits, however. 4. DISCUSSION
01 0
I
I
100
200
300
p,
STEP
DENSITY
I 400
500
600
( microns-‘)
FIG. 5. (a) Plot of trajectory slopes measured from the trajectories in Fig. (4) vs. step density. (b) Step flux vs. step density derived by integrating (a).
appropriate x-time trajectories were straight lines, but the slopes of the trajectories of a given p measured from the different pits were not the same. However, it was observed for a range of solution concentrations that a large increase in current density resulted in the pits losing their leading-edge discontinuity in that a region of shallow slope was generated at the pit peripheries. An example of this effect is shown in Fig. 9. The bromide concentration for this case was 0.10 M. In the first two frames the current density was 10 mA/cms. In the third and fourth frames the current density was increased to 30 mA/cms, and in the last two frames it was again 10 mA/cms. For the 30mA/cms condition the slope at the bottom of the pit became very shallow and illdefined. When the current density was returned to
Jr.
AND CONCLUSION
The results of this study are in accord generally with the continuum kinematic theory of dissolution. The prediction of linear x-time trajectories, along which p is constant, were observed for a variety of dissolution conditions, and the J-p curves derived have forms similar to those postulated by Cabrera and Frank. For shallow etch pits and other features with small values of p to be clearly defined they should have leading-edge discontinuities at their peripheries. The kinematic theory shows that there must be positive curvature in the J-p curves to maintain leading-edge discontinuities; and therefore Type II curves shouId characterize the more desirable etching conditions for delineating smalI-p features. This effect was also demonstrated by this study, for the peripheries of the etch pits developed under conditions corresponding to Type I curves had no leading-edge dis~ontinuities and were not easily seen under the microscope, while the shallow etch pits developed under conditions giving rise to Type II curves were much better defined. The shallow pit shown in Fig. l(b) was developed under conditions required for Type II curves. Note that the fringe spacings are smaller at the periphery, indicating a leading-edge discontinuity. An outstanding question concerning dissolution in the HCl-HBr-copper system is the role of bromide ion, the essential ingredient for developing easily seen etch pits. Figure 7 shows that the
APPLICATION
OF THE
l
OS03 M
KINEMATIC
THEORY
TO
COPFER
CRYSTAL
ETCHING
STUDIES
12%
HE3i
o 0,255 M HBr A i.0
M HBr
CURRENT DENSITY:
0
lcm
200
5 ma/cm2
300
400
500
600
700
800
p i steps/p) Frc. 7. Step flux vs. step density curves for snlutions of varying bromide ion cancentration,
increase in bromide concentration changed the mechanics of dissolution such that the J-p curve was transformed from Type I to Type II. Frank and Cabrera postulate that Type II curves result from ‘poisons’ whose adsorption on the surface is time dependent. One might speculate that bromide acts as a poison, but the data from this study are not sufficient for conclusions eancerning the atomistic processes. The formulation of the kinematic theory and Frank and Cabrera’s proof that Type II curves produce more salient topographical features do not presuppose any specific atomistic mechanisms. However, the rofe of bromide is at least defined phenome~olo~~ally in that, according to Fig. 7, it causes the dissolution to be governed by a Type II curve, Conditions giving rise to Type I curves cannot cause leading-edge discontinuities, but will permit formation of trailing-edge discontinuities. For a Type I curve, the derivative (c’Ji+)/p=pl continuously decreases as pl increases, so that areas of smafler slopes move faster than those with larger
slopes. In the evolution of the profile shown in Fig. 8, for which a Type I curve {same experiment as for Curve A in Fig. 7) applied, the lower slopes at the rear of the profile converged on the higher ones in the middle to form a trailing-edge discontinuity, while the lower slopes at the front of the profile diverged from those in the middle. We were not able to show quantitatively the effects of current density on the 3-p curves. AIthough the x-time trajectories were straight lines, the results were not consistent between the various pits for current densities larger than 10 mtlijcmz. A possible explanation for this result is that the changes in concentration overpotential associated with changing concentrations of copper and/or bromine ions vary among the different pits, and at the larger current densities this variation may serve to alter the trajectory slopes. However, the generation at large current densities of regions of shallow slope at the pit peripheries which moved much faster than regions of higher slope was suggestive of a Type I, J-p curve (see the preceding
1292
L.
D.
HULETT,
Jr.
paragraph). Also, this shallow pit periphery was a feature of the pits formed at 5 mA/cms in 0.03 M HBr, 6 M HCl solution, for which a Type I curve was demonstrated. Therefore, we suggest that an increase in current density tends to shift the J-p curve from Type II to Type I. The topography changes shown in Fig. 9 can be accounted for qualitatively if the current density increase did cause the J-p curve governing dissolution to change from Type II to Type I. For the 10 mA/cms condition the features on the surface were sharply defined, so dissolution was governed by a Type II curve; see sketch in Fig. 10. At 30 mA/cma (frames 3 and 4 in Fig. 9) the
and
F.
W.
YOUNG,
Jr.
shallow-slope regions at the periphery of the pit moved much faster than regions of higher slope. The 30 mA/cma J-p curve is therefore assumed to be Type I, see Fig. 10. Nucleation rate (the JO’S in Fig. 10) has the units of steps/set and is the boundary condition for the step tlux. Increasing the current density increases the nucleation rate; and, if there were no change in the J-p curve, a higher slope would be generated. But since the J-p curve also changes, it is possible that the increase in nucleation rate can be over-ridden in such a manner that a smaller slope results (see Fig. 10). Thus, the pit in Fig. 9 assumes a smaller slope at its bottom for the 30 mA/cms condition. REFERENCES
i TYPE
Jo
I )
(30)
J (TYPE
J”(lO)
p” (30)
p”
III
(101
FIG. 10. Schematic diagram of the effect of large changes in current density on the J-p curve.
1. FRANK F. C., Growth and Perfection of Crystals, (edited by DOREMUSR. H., ROBERTS B. W. and TURNBULL D.) p. 411. John Wiley and Sons, New York (1958). 2. CABRERAN., Reuctiuity of Solids, (edited by DEBOER J. H.) p. 345. Elsevier, Amsterdam (1961); Growth and Perfection of Crystals, (edited by DOREMUSR. H., ROBERTS B. W. and TURNBULL D.) p. 393. John Wiley and Sons, New York (1958). 3. CHERNOVA. A., Sov. Phys. Uspekhi 4, 116 (1961). 4. MULLINS W. W. and HIRTH J. P., J. Phys. Chem. Solids 24, 1391 (1963). 5. IVES M. B., J. Appl. Phys. 32, 1534 (1961). 6. FRANK F. C. and IVES M. B., J. Appl. Phys. 31, 1996 (1960). 7. BATTERMANB. W., J. Appl. Phys. 28, 1236 (1957). 8. YOUNG F. W., Jr. and Hu~rrrr L. D., Jr., Metal Surfaces, (edited by ROBERTSON W. D. and GJOSTEIN N. A.) p. 375. Am. Sot. Metals, Metals Park, Ohio (1963). 9. YOUNG F. W., Jr. and SAVAGEJ. R., J. Appl. Phys. 35, 1917 (1964).
Solids
Pergamon
Press 1965. Vol. 26, pp. 1287-1292.
AN APPLICATION
Printed in Great Britain.
OF THE KINEMATIC
COPPER CRYSTAL ETCHING L. D. HULETT, Solid State Division,
Jr. and
F. W. YOUNG,
Oak Ridge National Laboratory, (Receiued
1 February
THEORY
TO
STUDIES* Jr.
Oak Ridge, Tennessee
1965)
Abstract-The kinematic theory of crystal dissolution predicts the shape of a dissolving crystal as a function of time, and suggests a convenient technique for measuring reaction rate as a function of crystallographic orientation. It has been applied to a study of dislocation etch pit growth on (111) surfaces of copper undergoing anodic dissolution in solutions containing HCI and HBr. The dissolution shapes of these pits at successive times have been determined from interference photomicrographs taken as the reaction progressed. The evolutions of the shapes have been shown to be in accord with predictions of the theory, and reaction rates as functions of orientations near (111) have been determined for various conditions of dissolution.
1. INTRODUCTION
THE surface of a crystal may be described by closepacked terraces separated by steps of atomic height, and dissolution of such a surface occurs by the removal of atoms from the steps, resulting in the progression of these steps across the surface. The so-called kinematic theory of crystal dissolution (or growth) which was proposed by FRANK(~) and CABEERA@) describes the dissolution in terms of the flow of these steps. While CHERNOV(3) has shown that this theory can be applied also to non close-packed surfaces, we will be concerned with close-packed and vicinal surfaces and so will follow the earlier treatments. This kinematic theory of crystal dissolution has been used by CABRERA@) to describe the formation of etch pits at dislocation sites during dissolution. When the step flux J at a given point on the surface is a function only of the step density p at that point, the kinematic theory predicts that points of a given orientation (p) will have straight line trajectories as dissolution proceeds. If the observation of such straight line trajectories for a crystal undergoing dissolution indicates that the kinematic theory applies for that system, this elegant theory can * Research sponsored Commission Corporation.
under
by the U.S. Atomic Energy contract with the Union Carbide
then be used in the description and elucidation of the dissolution. Frank and Cabrera define step density as proportional to the slope of a surface (with respect to a close-packed plane), then treat the surface as if it were continuous. While MULLINS and HIRTH@) have recently reformulated the theory, taking into account the discontinuous nature of stepped surfaces, the present paper is concerned only with the continuum theory. Dissolution studies are greatly simplified for systems to which the kinematic theory is applicable. IVES@) first demonstrated the quantitative applicability of the theory in the dissolution of LiF, and IVES and FRANK@) also used it to treat BATTERMAN’S work(7) on the dissolution of germanium. The authors have studied the growth of dislocation etch pits on (111) surfaces of copper undergoing anodic dissolution in solutions containing HCl and HBr.@) In the present paper we make a more rigorous application of the kinematic theory of dissolution to this process in order to interpret the evolution of the topography of the pits. 2. EXPERIMENTAL
PROCEDURE
The specimens, prepared by the methods of YOUNG and SAVAGE,@) were highly-perfect (ZOO1287
1288
L.
D. HULETT,
Jr.
1000 dislocations/ems) single crystal wafers, 2.5 cm dia. x O-7 cm thick, of 99.999% pure copper. They were mounted in silicone rubber bases so that only the (111) orientation on the flat side of the wafer was exposed to the solution. Etching was done anodically in mixtures of HCl and HBr solutions. The experimental set-up, similar to that used by the authors in previous work,@) was such that the crystal surface could be observed with an optical microscope during etching. The growth of the etch pits was recorded by time-lapse photomicrography using an interference microscope and a 3.5 mm automatic camera. The concentration of HCl was always 6 M, the concentration of HBr was varied in the range 0.03 M to 1.0 M, and the current densities ranged from S-30 mA/cms.
3. RESULTS Figures l(a) and l(b) illustrate the two general types of dislocation pits that were observed. The pit shown in Fig. l(a), having steeper sides, was generated by a dislocation that nucleated steps faster than the dislocation that generated the pit seen in Fig. l(b). They are called deep and shallow pits, respectively. The deep pitswere used in studying the motion of vicinal surfaces 3-8” from (ill), while the shallower pits were used to study surfaces of 14-3” misorientation. No data could be taken on surfaces below IQ” misorientation. The montage in Fig. 2 illustrates the growth of a deep pit. Neither of the two types of pits had completely smooth sides, for there was always a certain amount of ledge structure. This structure, which was usually more apparent on the lower misoriented varied with dissolution conditions; surfaces, generally the average ledge height was N 300 A. However, for the purpose of this paper, the existence of ledges was ignored, and measured step densities represent average misorientations of the surface. Ledges will be discussed further in a later publication. Pit profiles in the (211) directions were constructed from interference photomicrographs as in Fig. 3(a). A family of curves of this type could be obtained from photomicrographs taken at successive times as the pit grew in size as dissolution proceeded. Differentiating the curve in Fig. 3(a) produced the p vs. x curve in Fig. 3(b), which
and F. W. YOUNG,
Jr.
represents the distribution of step density along the side of the pit. From a time-dependent family of curves of the type shown in Fig. 3(b), x vs. time trajectories could be determined for fixed values of P*
Figure 4 is a plot of the trajectories of various step densities measured from the two types of pits shown in Fig. 1 for a current density of 10 mA/cms in 6 M HCl-0.25 M HBr solution. The linearity of these trajectories indicate that the kinematic theory of dissolution is applicable, since Cabrera and Frank show that a necessary condition for the applicability of this theory is that x-time trajectories be linear. From the theory, the slope of the trajectory of a given step density pl is equal to (aJ/8p)jpl. Since J is a function of p only, the slope of the trajectory must be constant. From the slopes of the trajectories the J vs. p curve can be calculated : P
J
=
sSdp+J(po)
(1)
PO
where
=
slope of the trajectory
of a point of density p.
Figure 5(a) is a plot of the slopes of the trajectories shown in Fig. 4. By graphical integration of the curve in Fig. 5(a), the J vs. p curve in Fig. 5(b) was obtained. The integration constant, J(po), was determined from the motion of the discontinuity formed at the intersection of the bottom of a flat-bottomed pit with the pit side. Figure 6 is a photograph of such a flat-bottomed pit. The dislocation that caused this pit was moved from this site so that nucleation stopped, and as the pit sides continued to be attacked they left a wake of almost perfectly oriented (111) surface. The velocity of the discontinuity in the x-direction and the step density of the pit side that made up the discontinuity were measured simultaneously. From these measurements the step flux JO corresponding to the step density po of the pit side at the point of intersection was calculated. From the kinematic theory of
FIG. 1. Interference photomicrographs of deep pit (a) and shallow pit (b) generated at dislocation sites on (111) surface of copper. Dissolution conditions: 6 M HCl, 0.25 M HBr, 10 m,4/cm2, 50-set dissolution time. Magnification = 990 x .
500
steps/p
5 i*o
Y
1”
T ‘0 P
(a)
/
x
[ii21
FIG. 3. Construction
of the profile (a) of a dislocation pit and the derivation distribution (b).
of the step density
FIG.
0.
Interferrnce
photomicrograph of fat-bottomed 900 x Magnitication
pit.
FIG, 8. Evolution
of a treilinp-edge discontinuity Dissolution conditions:
in a pit growing under the conditions of a Type I step flux vs. step 6 M HCI, 0.03 I.%1HBr, 5 m.~/em2. .50+x frame intervals.
density
curve.
2
APPLICATION
OF THE
KINEMATIC
THEORY
TO
COPPER
CRYSTAL
310
100
ETCHING
270
STUDIES
1289
210
510 t-----
FIG. 4. X-time
dissolution by:
trajectories along which step density is constant. Step micron) associated with each trajectory is indicated.
the velocity of a discontinuity
V PO.Pl
$0) =
is given
- Jh)
PO-P1
(2)
where po and pl are the step densities of the two surfaces that make up the discontinuity, and J(p0) and J(pl) are the respective step fluxes. In the case of the discontinuity at the bottom of a flat-bottomed pit such as in Fig. 6, the J and p of the (111) surface are both zero. Then, J(Po) Vpo,o = PO From the experimentally determined values of Vp0.0 and po the value of J(po), the integration constant, was calculated. It has been found that the x-time trajectories along which p is constant are straight lines for current densities 5-20 mA/cms in solutions with bromide ion concentrations of 0.03-1.0 M. For a current density of 5 mA/cms, J vs. p curves for
density
(steps/
bromide ion concentrations of 0.03, O-25, and 1 *OM were determined, see Fig. 7. For small concentrations of bromide ion (@J/ap2) is negative for all p (curve A, Fig. 7), while for higher bromide ion concentration the curves have inflection points such that (SJ/ap2) is positive for small values of p (curves B and C). Following CABRERA@) we denote curve A as Type I, and curves B and C as Type II. The series of photographs and their corresponding p-x curves shown in Fig. 8 demonstrate the evolution of a trailing-edge discontinuity. An early variation in nucleation rate caused the profile of this pit to be sigmoid. After the second frame the nucleation rate had completely stopped, and the pit was flat-bottomed. The shallow slopes originally generated behind overtook and compromised the larger slopes in the mid-portion of the profile. This pit was grown under conditions corresponding to a Type I curve (curve A in Fig. 7). Note that a region of shallow slope was generated at the periphery of this pit. We have obtained some information on the effect of changing the current density, in a solution
L.
1290
D. HULETT,
Jr.
of given bromide concentration, on the J-p curves; compare the curve in Fig. 5(b), 10 mA~cm2, with curve 3 in Fig. 7,s mA/cms. But, for current densities higher than 10 mA/cms there were difficulties in determining the J-p curves. The too,
I
/
I
I
I
1
and
F.
W.
YOUNG,
10 mA/cms the higher, better defined slope was regenerated. When the pit was in the ‘flatbottomed’ state the dislocation that caused it had not stopped nucleating, as in the case of the pit in Fig. 8. In fact, the nucleation rate was higher at 30 mA/cma than at 10 mA/cms. Note also that a region of very shallow slope, which was propagated much faster than the remainder of the pit, was generated at the periphery when the current density was 30 mA/cma. When the current density was returned to 10 mA/cms, the shallow periphery stopped moving and eventually disappeared. The ‘flat-bottomed’ state of the pit was rather extraordinary. Only a smali percentage of the dislocation pits behaved this way. The slopes at the bottoms of most pits were increased by an increase in current density. The generation of the shallow periphery was observed for all pits, however. 4. DISCUSSION
01 0
I
I
100
200
300
p,
STEP
DENSITY
I 400
500
600
( microns-‘)
FIG. 5. (a) Plot of trajectory slopes measured from the trajectories in Fig. (4) vs. step density. (b) Step flux vs. step density derived by integrating (a).
appropriate x-time trajectories were straight lines, but the slopes of the trajectories of a given p measured from the different pits were not the same. However, it was observed for a range of solution concentrations that a large increase in current density resulted in the pits losing their leading-edge discontinuity in that a region of shallow slope was generated at the pit peripheries. An example of this effect is shown in Fig. 9. The bromide concentration for this case was 0.10 M. In the first two frames the current density was 10 mA/cms. In the third and fourth frames the current density was increased to 30 mA/cms, and in the last two frames it was again 10 mA/cms. For the 30mA/cms condition the slope at the bottom of the pit became very shallow and illdefined. When the current density was returned to
Jr.
AND CONCLUSION
The results of this study are in accord generally with the continuum kinematic theory of dissolution. The prediction of linear x-time trajectories, along which p is constant, were observed for a variety of dissolution conditions, and the J-p curves derived have forms similar to those postulated by Cabrera and Frank. For shallow etch pits and other features with small values of p to be clearly defined they should have leading-edge discontinuities at their peripheries. The kinematic theory shows that there must be positive curvature in the J-p curves to maintain leading-edge discontinuities; and therefore Type II curves shouId characterize the more desirable etching conditions for delineating smalI-p features. This effect was also demonstrated by this study, for the peripheries of the etch pits developed under conditions corresponding to Type I curves had no leading-edge dis~ontinuities and were not easily seen under the microscope, while the shallow etch pits developed under conditions giving rise to Type II curves were much better defined. The shallow pit shown in Fig. l(b) was developed under conditions required for Type II curves. Note that the fringe spacings are smaller at the periphery, indicating a leading-edge discontinuity. An outstanding question concerning dissolution in the HCl-HBr-copper system is the role of bromide ion, the essential ingredient for developing easily seen etch pits. Figure 7 shows that the
APPLICATION
OF THE
l
OS03 M
KINEMATIC
THEORY
TO
COPFER
CRYSTAL
ETCHING
STUDIES
12%
HE3i
o 0,255 M HBr A i.0
M HBr
CURRENT DENSITY:
0
lcm
200
5 ma/cm2
300
400
500
600
700
800
p i steps/p) Frc. 7. Step flux vs. step density curves for snlutions of varying bromide ion cancentration,
increase in bromide concentration changed the mechanics of dissolution such that the J-p curve was transformed from Type I to Type II. Frank and Cabrera postulate that Type II curves result from ‘poisons’ whose adsorption on the surface is time dependent. One might speculate that bromide acts as a poison, but the data from this study are not sufficient for conclusions eancerning the atomistic processes. The formulation of the kinematic theory and Frank and Cabrera’s proof that Type II curves produce more salient topographical features do not presuppose any specific atomistic mechanisms. However, the rofe of bromide is at least defined phenome~olo~~ally in that, according to Fig. 7, it causes the dissolution to be governed by a Type II curve, Conditions giving rise to Type I curves cannot cause leading-edge discontinuities, but will permit formation of trailing-edge discontinuities. For a Type I curve, the derivative (c’Ji+)/p=pl continuously decreases as pl increases, so that areas of smafler slopes move faster than those with larger
slopes. In the evolution of the profile shown in Fig. 8, for which a Type I curve {same experiment as for Curve A in Fig. 7) applied, the lower slopes at the rear of the profile converged on the higher ones in the middle to form a trailing-edge discontinuity, while the lower slopes at the front of the profile diverged from those in the middle. We were not able to show quantitatively the effects of current density on the 3-p curves. AIthough the x-time trajectories were straight lines, the results were not consistent between the various pits for current densities larger than 10 mtlijcmz. A possible explanation for this result is that the changes in concentration overpotential associated with changing concentrations of copper and/or bromine ions vary among the different pits, and at the larger current densities this variation may serve to alter the trajectory slopes. However, the generation at large current densities of regions of shallow slope at the pit peripheries which moved much faster than regions of higher slope was suggestive of a Type I, J-p curve (see the preceding
1292
L.
D.
HULETT,
Jr.
paragraph). Also, this shallow pit periphery was a feature of the pits formed at 5 mA/cms in 0.03 M HBr, 6 M HCl solution, for which a Type I curve was demonstrated. Therefore, we suggest that an increase in current density tends to shift the J-p curve from Type II to Type I. The topography changes shown in Fig. 9 can be accounted for qualitatively if the current density increase did cause the J-p curve governing dissolution to change from Type II to Type I. For the 10 mA/cms condition the features on the surface were sharply defined, so dissolution was governed by a Type II curve; see sketch in Fig. 10. At 30 mA/cma (frames 3 and 4 in Fig. 9) the
and
F.
W.
YOUNG,
Jr.
shallow-slope regions at the periphery of the pit moved much faster than regions of higher slope. The 30 mA/cma J-p curve is therefore assumed to be Type I, see Fig. 10. Nucleation rate (the JO’S in Fig. 10) has the units of steps/set and is the boundary condition for the step tlux. Increasing the current density increases the nucleation rate; and, if there were no change in the J-p curve, a higher slope would be generated. But since the J-p curve also changes, it is possible that the increase in nucleation rate can be over-ridden in such a manner that a smaller slope results (see Fig. 10). Thus, the pit in Fig. 9 assumes a smaller slope at its bottom for the 30 mA/cms condition. REFERENCES
i TYPE
Jo
I )
(30)
J (TYPE
J”(lO)
p” (30)
p”
III
(101
FIG. 10. Schematic diagram of the effect of large changes in current density on the J-p curve.
1. FRANK F. C., Growth and Perfection of Crystals, (edited by DOREMUSR. H., ROBERTS B. W. and TURNBULL D.) p. 411. John Wiley and Sons, New York (1958). 2. CABRERAN., Reuctiuity of Solids, (edited by DEBOER J. H.) p. 345. Elsevier, Amsterdam (1961); Growth and Perfection of Crystals, (edited by DOREMUSR. H., ROBERTS B. W. and TURNBULL D.) p. 393. John Wiley and Sons, New York (1958). 3. CHERNOVA. A., Sov. Phys. Uspekhi 4, 116 (1961). 4. MULLINS W. W. and HIRTH J. P., J. Phys. Chem. Solids 24, 1391 (1963). 5. IVES M. B., J. Appl. Phys. 32, 1534 (1961). 6. FRANK F. C. and IVES M. B., J. Appl. Phys. 31, 1996 (1960). 7. BATTERMANB. W., J. Appl. Phys. 28, 1236 (1957). 8. YOUNG F. W., Jr. and Hu~rrrr L. D., Jr., Metal Surfaces, (edited by ROBERTSON W. D. and GJOSTEIN N. A.) p. 375. Am. Sot. Metals, Metals Park, Ohio (1963). 9. YOUNG F. W., Jr. and SAVAGEJ. R., J. Appl. Phys. 35, 1917 (1964).