# Metal Assisted Chemical Etching patterns at a Ge/Cr/Au interface
modulated by the Euler instability

Yilin Wong  Giovanni Zocchi<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of California - Los Angeles

###### Abstract

We present a solid state system which spontaneously generates remarkable
engraving patterns on the surface of Ge. The layered construction, with a
metal film on the Ge surface, results in coupling of the metal catalyzed
etching reaction with the long range stress field at the Ge - metal interface.
The etching patterns generated have similarities with Turing patterns,
hydrodynamic patterns, crack propagation, and biological form. We describe
spirals, radial patterns, and more disordered structures. Euler buckling of
the metal layer generates a characteristic wavelength for some patterns.

Keywords Metal Assisted Chemical Etching, pattern formation, reaction
diffusion systems, Euler instability, mechano - chemistry, crack propagation,
screw dislocation, corrosion.

##  I Introduction

Understanding patterns in nature is a basic quest of science [1]. While
biological form [2] inspired the pioneering work of Alan Turing [3] on
reaction - diffusion systems, most nonlinear dynamics studies that followed
focussed on simpler experimental systems, such as thermal convection or
crystal growth [4, 5, 6]. Here we present a solid state system which, starting
from homogeneous initial conditions, generates remarkable patterns at the
$10-100\,\mu m$ scale. The patterns arise from a complex interplay of solid
state mechanics, chemical reactions, and diffusion. Although the system is
very simple, its behavior is complex. It intersects several topics in
nonlinear physics and engineering. The basis for the patterns is Metal
Assisted Chemical Etching (MACE [7]), combined with thin film mechanical
instability [8].
Initially shown to produce porous Si for optoelectronics applications [9],
MACE is a relatively recent technique pursued for high throughput fabrication
of 3D structures on semiconductor surfaces [10, 11, 7]. It consists of
depositing a noble metal, in the form of nano particles or a thin film, onto
the surface of Si or Ge. In the presence of oxidizing agents such as $HF$ or
$H_{2}O_{2}$ the metal catalyzes etching of the semiconductor, creating 3D
patterns such as deep wells [10]. By shaping the metal catalyst, it is
possible to design specific etching patterns, such as spirals [12]. These
applications take advantage of the metal catalyst to localize and direct the
etching reaction. 3D structures can be obtained because the metal catalyst,
e.g. in the form of nano particles, “travels” with the etching front [13].
The buckling of thin metal films under compressive stress has been studied
extensively in the past few decades due to its relevance to diverse fields
such as flexible electronics [14], design of structural sandwich panels [15]
and the understanding of mechanical instabilities in morphogenesis [16].
Understanding the instabilities of thin films under tensile stress is equally
important, with recent work highlighting different regimes of the wrinkling
instability [17] and addressing patterns formed by crack propagation in the
film [18].
Our system is prepared from a p-doped, $\sim 200\,\mu m$ thick Ge wafer (see
Mat. & Met. in Supp. Mat. for details). A $\sim 10\,nm$ thick Cr layer is
evaporated on the (100) surface of the chip, followed by a $4\,nm$ Au
evaporation. To start the process, a drop of mild etching solution is
transferred on the metallized surface and allowed to dry overnight. After
washing off precipitated salt, the chip is re-incubated with the same etching
solution in a wet chamber to prevent evaporation. From these homogeneous
initial conditions, in the course of 24 - 48 hrs, remarkable patterns emerge
on the Ge surface, revealed by the metal layer lifting off. The long range
elastic stress field in the metal film couples with catalysis at the Ge -
metal interface and diffusion of the etching solution to provide a mechanism
for instability which can result in ordered large scale patterns. With coupled
local (diffusion and reaction controlled) and global (elastic field
controlled) growth mechanisms, the patterns obtained have a Turing - like as
well as a hydrodynamic character. The thickness of the metal layer, the
initial state of mechanical stress of the sample, and the composition of the
etching solution all play a role in determining the type of pattern that
develops.

##  II Experimental Observations

The system is prepared as a laterally homogeneous solid state construct (i.e.
with no pre-imposed pattern) consisting of the Ge substrate, an evaporated
layer of Cr, and a further layer of Au (Mat. & Met.). Then a mild etching
solution, basically slightly acidic water ($H_{2}KPO_{4}$ buffer at $pH=4$),
is added. Over a period of $\sim 24-48$ hrs, a variety of stunning geometrical
surface patterns at the $\sim 100\,\mu m$ scale appear. These patterns are
etched onto the Ge surface, at the interface with the metal film; the latter
eventually lifts off the Ge surface during the etching process. For given
conditions, one pattern, say spirals, is dominant or even exclusively present,
appearing at seemingly random locations, so that on a chip $\sim 1\,cm^{2}$ in
size there may be hundreds of nearly identical spirals. There appear to be two
“fundamental” geometric patterns: the spiral (Fig. 1) and the radial pattern
(Fig. 2). The spirals are further subdivided into Archimedean (Fig. 1(a)) and
Logarithmic (Fig. 1(b)). We can also generate other, seemingly disordered
patterns (Fig 3): we believe these correspond to conditions “in between” those
giving rise to either spirals or radial patterns, possibly also in the
presence of gradients in the system (e.g. temperature gradients). The basic
patterns always originate from a central “etch pit” and grow outwards. Fig. 1
shows pictures obtained by reflection optical microscopy of the dry sample;
the square structure at the center of the spiral is the etch pit. It has an
inverted pyramidal structure with flat bottom and terraces, and aspect ratio
$\sim 1$. The lateral size is $2-10\,\mu m$ in our samples, with similar
depth. The sides of the square are always alligned with the lattice axes of
the (cubic) Ge crystal, as we see from the fact that all etch pits are
oriented the same way on the [100] surface of the Ge wafer. Fig. S5 (Supp.
Mat.) shows an SEM picture of the etch pit. From the etch pit, $2-6$ “arms”
may sprout out, typically at the corners of the square. In Fig. 1(a), the
terraced square structure of the pit “sprouts” two arms from the NE corner;
these become the ridges of the two-armed tight spiral in the figure. Note
that, contrary to appearances, the pattern in the figure forms a valley, not a
hill, on the Ge surface. As one walks (clockwise) along one or the other of
the two spiral terraces, moving outwards from the center, one is “climbing”
towards the level of the unetched surface of the chip. Indeed the maximum
diameter of the pattern ($\sim 200\,\mu m$ in Fig. 1(a)) corresponds to the
spiral terraces reaching the level of the unperturbed Ge surface. The step
separating one terrace from the adjacent one is of order $1\,\mu m$. In Fig.
1(a) we count 10 terraces in the radial direction; the overall depth change,
measured approximately by focussing the microscope alternately near the center
and at the rim, is approximately $10\,\mu m$. In the picture, the metal film
has lifted off the Ge surface. In general, as the etching reaction proceeds,
the metal film corrugates and eventually lifts off the Ge surface. In the
spiral of Fig. 1(a), the distance between the two arms (i.e. the width of the
terraces) is constant as one moves radially outwards. Geometrically this
corresponds to a curve, in polar coordinates $(r,\theta)$, given
parametrically by $\\{r=ut\,,\,\theta=\omega t\,,\,0\leq t\leq t_{max}\\}$
(Archimedean spiral). The distance between the same arm at two successive
turns is $\Delta r=2\pi u/\omega$ , independent of $r$. In Fig. 1(a), this
distance is twice the width of one terrace. It is indeed constant in the
experiments, and also approximately the same across spirals on the same
sample.

(a)

(b)

Figure 1: (a) Archimedean spiral formed on Ge surface with $10\,nm$ Cr and
$4\,nm$ Au depositions ($1M\,KH_{2}PO_{4}$ buffer at $pH\,4$ with $10\,\mu M$
DNA). The square etch pit at the center is $9\,\mu m$ in lateral size and
$\sim 10\,\mu m$ deep. Two etch ridges (spiral “arms”) sprout from the NE
corner of the etch pit, forming a system of two intertwined ascending spiral
terraces. The width of a terrace is $10\,\mu m$ and the outer diameter of the
structure is $200\,\mu m$.
(b) Logarithmic spiral formed with $6\,nm$ Cr and $4\,nm$ Au on Ge, same etch
solution as in (a). The pattern is $600\,\mu m$ in diameter. The diagonal line
across the NW corner of the picture is a rolled up piece of metal film
detached from the Ge base.

The pattern of Fig. 1(b) is obtained with only a slight change in parameters
(thinner Cr layer). It has the same topology as the pattern in Fig. 1(a): the
two “arms” of the spiral are ridges delimiting two terraces which are sloping
upwards as one turns clockwise, moving outwards from the central pit. However,
the arms now follow an approximately logarithmic spiral shape (Supp. Mat. Fig.
S3), the width of the terraces increasing moving outwards. The pattern is more
jagged compared to the smoother Archimedean spiral, and the ridges are deeper.
In polar coordinates, the parametric form of the logarithmic spiral is
$\\{dr/dt=r/\tau\,,\,\theta=\omega t\,,\,0\leq t\leq t_{max}\\}$ , i.e.
$r=r_{0}\,e^{\theta/(\omega\tau)}$ . For the pattern of Fig. 1(b) ,
$\omega\tau\approx 3.8$ (Supp. Mat. Fig. S3).
Since etching involves primarily a redox reaction (see Discussion), we guessed
that addition of an oxidizing or reducing agent could alter the dynamics and
the patterns obtained. Indeed, Fig. 2 shows a quite different, radial pattern
obtained with the addition of TCEP (a reducing agent). The figure is an
optical microscope picture of the dry sample; the metal layer has lifted off
the area of the circular disk, exposing the Ge surface. The pattern of grooves
(darker lines) has on average a constant wavelength $\ell$ (distance between
grooves), maintained by “tip splitting” of the grooves as the pattern grows
radially. For the pattern of Fig. 2 , $\ell\approx 22\,\mu m$ , while the
diameter of the disk is $\approx 2\,mm$ . Like the spirals of Fig. 1 , the
radial pattern also starts from an etch pit at the center. Immediately
surrounding the pit there is a circular pattern, which then transitions to the
radial pattern (Fig. 2).

Figure 2: Radial pattern of grooves (dark lines) formed on Ge with $20\,nm$ Cr
and $4\,nm$ Au ($10\,mM$ TCEP, $KH_{2}PO_{4}$ buffer, no DNA). The metal layer
has lifted off the surface of the disk, which is $2\,mm$ in diameter. Tip
splitting maintains a constant wavelength of the pattern (distance between
grooves) of $\ell\approx 20\,\mu m$ .

Through time-lapse video microscopy of samples under water we have obtained
preliminary measurements of the growth dynamics of the basic patterns. Both
the Archimedean spiral and the radial pattern have constant radial growth rate
$u$ , of approximately $u=dr/dt\approx 0.75\,\mu m/min$ for the sample
measured (Supp. Mat. Figs. S6, S7). The Archimedean spiral “turns” at constant
angular velocity $\omega$ of approximately $6\,^{\circ}/min$ (Fig. S9) ; this
gives a wavelength of $2\pi u/\omega\approx 45\,\mu m$. Finally, besides the
regular spiral and radial patterns, more disordered - and quite arresting -
patterns can be obtained, under conditions which we do not entirely control
(Fig. 3). These appear to be combinations of radial and spiral elements,
possibly indicating that conditions are close to a bifurcation point
separating two different regular patterns.

Figure 3: “Scary patterns” formed on Ge with $3\,nm$ Cr and $4\,nm$ Au. Figure
4: Interference microscopy picture of a radial structure in water, before the
metal film lifts off. Light reflection from the Ge surface and the metal layer
forms the colorful interference fringes showing the corrugations of the metal
layer. The process gives rise to a pattern of grooves on the Ge as in Fig. 2.
The diameter of the disk is $22\,\mu m$.

##  III Discussion

We observe a stunning variety of patterns produced by metal catalyzed etching
of the Ge (100) surface. The patterns always start from an initial etch pit
and propagate outwards. The etch pit and surrounding tight spiral (Fig. 1a)
are well known patterns, thought to be associated with preferential etching at
a screw dislocation [19]. The association of etch pits with screw dislocations
is well established, indeed the former have been used to measure density and
dynamics of the latter [20]. The mechanism of formation of the surrounding
spiral is less established; several authors note that if etching of the spiral
was a time reversed crystal growth process along the screw dislocation, the
dislocation in question would have an enormous ($\mu m$ size) Burgers vector,
which seems implausible [19]. For our system, the observation that a variety
of different patterns can be obtained with slight changes in the parameters
points to other mechanisms controlling the growth. In particular, the spiral
patterns are reminiscent of those observed in some liquid phase reaction-
diffusion systems, such as the Belousov-Zhabotinsky reaction [21, 22, 23]. In
these systems, the instability is driven by an autocatalytic reaction. But
spirals, and the zig-zag and oscillatory patterns visible in Fig. 3, are also
characteristic of crack propagation in thin films [18, 24, 25]. Our patterns
similarly arise spontaneously from a nonlinear reaction - diffusion type
mechanism. However in our case, the growth mechanism involves the interplay of
chemical etching and mechanical stress, which deforms the metal film catalyst.
The Ge - Cr interface may be pre-stressed for several reasons, including
incommensurate lattice constants of Ge and Cr, and different thermal expansion
coefficients of Ge and the Cr film. Assuming the metal film is under
compressive stress, if the Ge surface is etched away from under it, the metal
film may buckle, leaving a void into which etching solution and products can
diffuse. The buckling instability of the metal film may be the mechanism which
selects the wavelength of the radial pattern, as we discuss below. Conversely,
if the metal film is under tensile stress, it may crack, facilitating
diffusion of the etch solution through the crack.
Etching of Ge proceeds through the formation of Germanium oxide ($GeO_{2}$).
In our system, a possible sequence of overall reactions is [11]:

$Ge+2H_{2}O\rightarrow GeO_{2}+4H^{+}+4e^{-}$ (1)

at the Ge - metal - water contact line, and

$O_{2}+4e^{-}+4H^{+}\rightarrow 2H_{2}O$ (2)

at the metal - water interface, for a net reaction $Ge+O_{2}\rightarrow
GeO_{2}$ . Ge oxide is somewhat soluble in water, according to
$GeO_{2}+2H_{2}O\rightarrow Ge(OH)_{4}$ . Note that while this is a possible
“overall” reaction sequence, the actual reaction steps may be more. Following
the Ge oxidation reaction (1) the metal layer may play the role of an electron
transporter, delivering electrons to the solution for the oxygen reduction
reaction (2). The Gold islands, which are necessary to generate the patterns,
may provide a conductive path across the oxide layer which likely forms at the
bare Cr - water interface, as well as providing a catalytic surface for the
reaction (2). Mechanisms of charge and mass transport are discussed in [26].
Our patterns are associated with the buckling of the metal layer following its
local detachment (“delamination”) from the Ge surface. Fig. 4 is an
interference microscopy picture of a radial pattern before the metal film
lifts off the Ge surface. The interference fringes created by reflection from
the semitransparent metal layer and the Ge surface show the buckling of the
metal film; this radial pattern results in the etching pattern on the Ge
surface shown in Fig. 2. The tip splitting process of Fig. 2 is also visible
in this interferogram.
Etching removes material from the Ge surface, creating a delamination area.
Buckling of the metal inhibits further etching of the delaminated area (since
in the void the metal catalyst is removed from the Ge surface), except at the
Ge - metal - liquid contact line, where etching is catalyzed. Depending on the
balance of forces on it, the contact line may move laterally, but also into
the Ge surface [13], creating grooves. Buckling also facilitates diffusion of
reactants to and from the contact line, and thus affects the reaction speed.
The pattern of Fig. 2 exhibits a characteristic wavelength which is maintained
by tip splitting as the pattern grows radially (Supp. Mat. Fig. S1). Assuming
that the observed wavelength $\ell$ corresponds to the critical point of the
Euler buckling instability for the metal layer, we can calculate the original
strain in the metal. For a thin film of thickness $h$ under uniaxial
compressive stress, the relation between critical strain $\epsilon$ and
critical length for buckling $\ell$ is approximately (Supp. Mat.):

$\epsilon=\frac{2\pi^{2}}{3}\frac{h^{2}}{(1-\nu^{2})\,\ell^{2}}$ (3)

where $\nu$ is the Poisson ratio. For Cr, $\nu\approx 0.2$ so
$(1-\nu^{2})\approx 1$ ; with $h=20\,nm$ (metal film thickness) and
$\ell=20\,\mu m$ (observed wavelength) we obtain $\epsilon\approx 6.6\times
10^{-6}$ for the strain. This level of strain could easily result from the
different thermal expansion of Ge and the metal film. The thermal expansion
coefficient for Ge is $\alpha_{Ge}\approx 6\times 10^{-6}\,{}^{\circ}K^{-1}$ ,
while for (bulk) Cr it is in the range $\alpha_{Cr}\approx 4.5-5.5\times
10^{-6}\,{}^{\circ}K^{-1}$ [27, 28]. As an order of magnitude, the mismatch in
thermal expansion coefficients in our system is therefore $\Delta\alpha\sim
10^{-6}\,{}^{\circ}K^{-1}$. The strain measured from the buckling instability
$\epsilon\approx 7\times 10^{-6}$ would then correspond to a temperature
difference $\Delta T=\epsilon/\Delta\alpha\sim 7\,^{\circ}C$. The sample of
Fig. 4 was incubated at $4\,^{\circ}C$, other samples at room temperature but
without controlling the temperature. Thus the metal layer may be under
compressive stress for some samples and under tensile stress for others.
In conclusion, we have presented a new pattern forming system which produces
remarkable structures. While some of the patterns are reminiscent of reaction
- diffusion systems, others recall patterns of crack propagation, or even
thermal convection. Then there are similarities with forms in the living
world. But this system is essentially solid state. Here, the mechanism for
instability and pattern formation involves the interaction of a catalytic
reaction with the mechanical deformation of the catalyst (the metal film).
This mechanism has been exploited in some applications of MACE, by designing
catalyst patches which fold into 3D shapes during the etching process [13],
and by designing 2D catalyst shapes that induce chiral etching patterns [12].
It is interesting to note that, on a different scale, mechano - chemical
coupling is also the basis for the workings of enzymes, the molecular machines
of life [29]. Coincidentally, the overall reaction (1), (2) parallels the
overall reaction of aerobic respiration, with Germanium in the role of
carbohydrates ! A theory for our patterns must describe the dynamics of the Ge
- metal - liquid contact line, a challenging free boundary problem for future
study. Similarly, identifying the experimental parameters which control growth
and morphology of the patterns is a challenge for future experiments. Finally,
the intersection with technologically important problems in materials science,
such as device fabrication, but also corrosion and crack propagation, surely
merits continued attention.

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