Supporting information:
Achieving reversibility of ultra-high mechanical stress by hydrogen loading of thin films
M. Hamm, V. Burlaka, S. Wagner, A. Pundt
Linear elastic theory on hydrogen-loaded thin Nb films clamped to a rigid substrate:
By interstitial solution of hydrogen, the metal lattice expands by o. This expansion scales
linear on the hydrogen concentration cH, with o = α × cH 1. However, depending on the
clamping condition, the lattice is forced to stay in its original size, for the clamped parts. By
this, mechanical stress builds up. A model comparing the situation for a free film (a) and a
film clamped to a substrate (b), is shown in Fig. 1. The clamped film can be regarded as a free
film that is compressed by o in the lateral directions, as shown in Fig. 1 (b)
Fig.1.: Scheme to explain strain and stress appearing in M-H thin films, as calculated by using linear elastic
theory. (a) A free film expands by 0 in all three room direction upon hydrogen uptake, (b) the film clamped to a
rigid substrate is forced to stay in its original size in the adhered in-plane xy area. Thereby, mechanical stress
arises in the film and an additional strain Δzz occurs in the out-of plane z-direction.
Applying the theory of linear elasticity yields a linear elastic stress increase and a vertical
strain increase upon hydrogen uptake 1,2. The in-plane stresses σii, for i=1 and 2, are linked to
the suppressed in-plane expansion o via the direction dependent biaxial modulus M with its
components Mijkl by
 ii  M iikk  ( 0 ) .
(1)
In out-of-plane direction, the stress is zero.
z  0
(2)
In this direction the film strains with o + Δε 3,4 with Δε resulting from transverse contraction.
For the (1-10)-oriented cubic single crystalline film this gives
 [110]
 (C  3C12  2C44 ) 
 0
 1  11
(C12  C11  2C44 ) 

(3)
and in-plane
 [110]   4
(C11  2C12 )C44
0
C12  C11  2C44
(4)
4C C  C  2C12  C11 (C12  2C 44 )
 [001]   12 44 11
0
C12  C11  2C 44
2
2
(5)
with Cij as bulk elastic constants of the cubic crystal, in Voigts notation. Note that the stress
differs in the two perpendicular in-plane directions [110] and [001]. With αH = 0.058 for H in
Nb, or ε0=0.058 × cH 1 and with C11=245.6 GPa, C44=29.3 GPa and C12=138.7 GPa, at 300 K
5
Eq. (3)-(5) yield ε[110] = 0.137 × cH, σ[110]= -7.6 [GPa] ×cH , σ[001]= -11.6 [GPa] × cH. These
are the two extreme values and the stress curve is sinoidal. In the film direction studied here
this gives σ[111]= -9.1 [GPa] × cH.
Thus, upon hydrogen loading to cH=1 H/Nb, ultra-high stress of about – 10 GPa is expected to
arise in the linear elastic regime.
Plastic deformation at the film/substrate interface
For epitaxial film growth on substrates, Mathews and Blakeslee
6-8
, van der Merwe
9,10
, or
Jain et al.11 give a critical yield thickness df* above which dislocations are inserted at the
film/substrate interface to accommodate for the misfit strain  between the film and the
substrate. The total energy density of the system tot mainly contains the elastic energy density
 el  M  d f   2 that builds up during film growth and the self energy density of edge
dislocations  disl 
Gb 2
2  d 
 which are introduced in periodic distances L. G is the
ln 
4 (1   ) L  rc 
shear modulus linked to E via E  2(1   )G , ν is Poisson’s ratio, b is Burgers vector, and rc
is the radius of the dislocation core which is often set to rc=b.  is a constant in the order of
one.
b
Gb 2
2  d


 M  d f     
ln 

L
4

(
1


)
L
r


 c 
(6)
2
 tot
Energy minimization gives a critical thickness, above which the first dislocation is introduced
and thus b/L=0, in equilibrium:
  d *f
1
d 
ln 
 8 (1   )  rc
*
f
b




(7)
This theory was recently extended to M-H films whereby lattice is strained by hydrogen by ε
= αH × cH 1. The onset of plastic deformation now depends on the hydrogen concentration
cHpl for each film of thickness d 3,12,18.
c H , pl 
 d f
b
1
ln 
 H  d f 8 (1   )  rc

1
  C1 
ln C2  d f

d
f


(7)
For the Nb-H system, the Burgers vector is of type b=1/2 {111}, and thus, with a = 3.30 Å 13,
|𝑏| = 2.86 Å. With rc = 2.86 Å, and ν=0.355 2,14, this theory predicts C1 = 1.4 ∙ 10-1 and C2 =
3.5 , in equilibrium.
The fit to our experimental data yields C1 = 1.2 and C2 = 5. While C2 matches the theory quite
well, C1 is about a factor of 9 larger than what is predicted above. Actually, we interpret this
discrepancy by the use of only a fraction of the available elastic energy.
Surface quality of epitaxial Nb films:
The topography of the film surface was studied by UHV-STM on Nb-films capped with less
than one monolayer of Pd, grown under comparable conditions. A very smooth Nb-film
surface topography with only a minor amount of screw dislocations was verified for these
film thicknesses. The average RMS roughness is 0.25 nm. Fig. 2 exemplarily shows the
surface topography of an 8 nm Nb film and a 25 nm Nb film. No pit-holes can be detected.
The mean terrace size is about 25 nm (RMS 0.27 nm) for the 8 nm film, and about 40 nm
(RMS 0.24 nm) for the 25 nm film. On top of this larger scale surface topography, tiny Pd-
islands are visible. The line scan reflects the surface topography as determined for 100 nm.
Height changes of 0.55 nm in maximum border the Nb-terraces and the monolayer Pd islands
modulation can be seen on top of them.
Fig.2 Surface topography of (a) an 8 nm and (b) a 25 nm Nb films, respectively. Surface terraces are smaller for
the 8 nm film. The surface is covered with less than a monolayer, visible as tiny islands on top of the larger scale
surface topography of the underlying Nb-film. A 100 nm line scan is provided, below the figures. It shows the
height changes being in the order of 0.55 nm, for both films. The surface RMS roughness is (a) 0.27 nm and (b)
0.24 nm.
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3
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