SUPPLEMENTAL MATERIAL
Structural models:
The (1x1) tetragonal (001) interface was built by a monolayer of B1-SiN sandwiched between five
layers of B1-ZrN. The (1x1) hexagonal (111) interface was constructed from a monolayer of Si
between two layers of N while the B1-ZrN along the [111] was modeled by 15 alternating Zr and N
monolayers. The (2x2) structures were derived by the in-plane periodic repetition of the (1x1)(see Fig.
S1).
Phonon calculations:
Calculations of phonons were performed on smaller structures: with a total of 12 atoms, whereof 1 Si,
5 Zr, and 6 N for the 001 interface, and 18 atoms, whereof 1 Si, 8 Zr, and 9 N for the (111) interface.
First step of the calculations were structural relaxations, any such calculations have to be initiated with
a small distortion of the atomic positions, otherwise the symmetry means that forces are cancelled out
and no relaxations occur. With the 001 interface the Si-atom moved about 14% away from the ideal
lattice position relative to total cell size before stabilizing. Phonon spectra on both ideal crystal
structure and relaxed structure shows the profound effect this relaxation had on the dynamic stability,
see Fig. S2. Note that in this case, since there is only one Si-atom in the structure there is no vacancy
and compared to Fig. 3a in the article all the Si-atoms would have distorted in the same direction.
With the (111) interfaces, relaxations were performed until they terminated at fulfillment of
convergence criteria (see below for details), but both with ideal coordinates and “relaxed” coordinates
the phonon spectra showed imaginary phonon frequencies in all directions, Fig. S3, suggesting a
distinct failure in obtaining a proper relaxed state. Figure 3b and d in the article show extreme and
irregular distortions, highly dependent on initial parameters, including volume, which we state is an
indication of dynamic instability in accordance with the following arguments.
A set of calculations intended to visualize the energy surface of the (111) interface was done by
performing repeated calculations while moving the Si-atom across the entire x-y plane of a (1x1)
supercell while within each calculation, all the atoms were kept fixed. The z-coordinate was chosen as
the ideal lattice position, which other calculations had shown to be stable for a given x-y position.
Since there are no relaxations involved and only a single Si-atom is controlled the problem could be
reduced to only 2 degrees of freedom allowing the surface to be easily visualized (this is of course a
simplification). As seen in Fig. S4, the ideal lattice positions are locally unstable with a circular
minimum surrounding a local maxima. When the Si-atom is allowed to follow a relaxation path it
could end up almost anywhere in that circular valley.
Given the results and conclusions it is also worth reminding the reader of a few points about how the
relaxations of atomic configuration works. In the calculations one determines net forces acting on the
atoms and based on those one moves the atoms a small distance within the cell to a new configuration
and then repeats the calculation of energy and forces in this new configuration. This process continues
until either the forces or the change in energy between two iterations fall below a certain predefined
tolerance level. Since there is, in any stable physical system, always a global energy minima (no
configuration will ever give an infinite negative energy), a point where the energy will not decrease
further and the forces on the atoms are zero. At such a point the ionic relaxations will terminate as the
convergence criteria is fulfilled. However, since we deal with a more or less flat energy surface in one
or more of the relevant degrees of freedom, in this case the iterations to determine the relaxed atomic
positions will terminate when the forces are sufficiently close to zero. But in order to have a
dynamically stable system it is vitally important that the atoms are relaxed into a stable energy
minimum in all degrees of freedom handled by the computational system. This can only be judged by
calculations of the phonon dispersion relations. But the numerical aspects discussed above show that
during the static calculations it is possible to complete relaxation of atoms into a state of the system
that turns out to be not stable dynamically.
Figure Captions:
Figure S1: Figures of the atomic structure model.(a) Ideal (001) interface ,(b) Ideal (111) interface.
Figure S2: Phonon spectra and related relaxations for 001 interface. top left) Energy surface as the Si
atom is moved to a fixed position relative to ideal lattice point (bottom left corner at coordinates (0,0))
and surrounding atoms are allowed to relax their positions relative to that. As can be seen, the ideal
lattice point is highly unstable and any distortion away from it would result in a reduced total energy.
top right) Atomic structure in interface layer in this representation, green is Si, blue is N, empty circles
are ideal positions and filled are final positions after relaxation. bottom left) Phonon spectra in ideal
crystal structure, there are imaginary phonon frequencies along the entire spectra indicating extreme
dynamical instability. bottom right) Phonon spectra after relaxation. The imaginary frequencies are
gone and the system has stabilized.
Figure S3: Phonon spectra for the (111) interfaces, ideal atomic coordinates (left) and “relaxed”
(right). In both cases there are imaginary frequencies along the entire spectra showing extreme
dynamical instability.
Figure S4: Energy surface for a Si-atom moving within the x-y plane of a (1x1) supercell of the (111)
ZrN/SiN interface. The ideal lattice position of the atom is at point x=0.333; y=0.667 at the top of the
smaller energy maximum. This maximum is surrounded by a ring that represents minima along which
there is little difference in energy. The atom would not be able to stabilize dynamically into any
position. It would always be able to move, more or less freely along this region.