Supplementary Information
This document provides additional graphs and tables which are not required to follow the reasoning in the main article.
A. Structural Information
TABLE S1. Coordinates for HfO2, HZO and ZrO2 of the f-phase. In HZO M1 and M2 are Hf, M3 and M4 are Zr. The assignment of
elements is the same in all other phases
HfO2
x
b = 5.04
HZO
ZrO2
y
z
x
y
z
x
y
z
a = 5.23
c = 5.06
b = 5.04
a = 5.24
c = 5.06
b = 5.04
a = 5.25
c = 5.06
M1
0.2668
0.0316
0.2565
0.2692
0.0316
0.2554
0.2679
0.0310
0.2522
M2
0.2668
0.4684
0.7565
0.2692
0.4684
0.7554
0.2679
0.4690
0.7522
M3
0.7332
0.5316
0.2565
0.7346
0.5308
0.2535
0.7321
0.5310
0.2522
M4
0.7332
0.9684
0.7565
0.7346
0.9692
0.7535
0.7321
0.9690
0.7522
O1
0.0680
0.3639
0.1119
0.0706
0.3647
0.1114
0.0726
0.3691
0.1174
O2
0.0680
0.1361
0.6119
0.0706
0.1353
0.6114
0.0726
0.1309
0.6174
O3
0.9320
0.8639
0.1119
0.9306
0.8676
0.1170
0.9274
0.8691
0.1174
O4
0.9320
0.6361
0.6119
0.9306
0.6324
0.6170
0.9274
0.6309
0.6174
O5
0.5372
0.2671
0.5076
0.5358
0.2669
0.5059
0.5394
0.2662
0.5064
O6
0.5372
0.2329
0.0076
0.5358
0.2331
0.0059
0.5394
0.2338
0.0064
O7
0.4628
0.7671
0.5076
0.4592
0.7666
0.5088
0.4606
0.7662
0.5064
O8
0.4628
0.7329
0.0076
0.4592
0.7334
0.0088
0.4606
0.7338
0.0064
TABLE S2. A comparison of computed and experimental cell volume and cell area ratio is stated. Experimental values according to
TABLE II and IV of the main article.
V0 = Vm
V0/Vf
V0/Vt
V0/Vo
V0/Vc
HfO2
ZrO2
HfO2
ZrO2
(*)
m-phase
f-phase
t-phase
o-phase
c-phase
exp.
1.000
tbd
0.965
0.961
0.951
comp.(*)
1.000
0.965
0.948
1.003
0.928
exp.
1.000
0.965
0.957
0.962
0.957
comp.(*)
1.000
0.965
0.951
1.001
0.932
A0 = Axz,m
A0/Axz,f
A0/Axz,t
A0/Axz,o
A0/Axz,c
m-phase
f-phase
t-phase
o-phase
c-phase
exp.
1.000
tbd
0.959
0.981
0.966
comp.(*)
1.000
0.956
0.937
0.989
0.937
exp.
1.000
0.957
0.959
0.981
0.970
comp.(*)
1.000
0.963
0.944
0.993
0.937
Computed values
B. Phonon Modes
FIG. S1. (a) is the computed IR active phonon intensity of the f-phase for HfO2, HZO, ZrO2; (b) and (c) the computed IR active
phonon intensity for the m-phase, o-phase, f-phase and t-phase. The distributions highly depend on the quality of the
pseudopotentials.
C. Bulk moduli
TABLE S3. Computed and experimental values for the bulk modulus K0, obtained from a Birch-Murnaghan fit to data
with fixed value of K0’=4. (*) own computations.
HfO2 in GPa
comp.
exp.
transition
pressure for
m- to o-phase
m-phase
o-phase
8(*)
41
198(*), 1681
246(*), 2181
1851
2661
f-phase
HZO in GPa
comp.
exp.
ZrO2 in GPa
comp.
exp.
12(*)
4.62)
185(*)
199(*)
155(*), 1433
171(*), 1953
2102)
2902)
229(*), 2724
224(*)
218(*), 2644
t-phase
210(*)
210(*)
205(*), 2143
c-phase
258(*), 2802
268(*)
257(*), 2674
D. Stress and strain
FIG. S2. (a) and (b) show the calculated ΔU for all phases of HfO2 for strain in xz- and yz-plain respectively. (c) and (d) show the
calculated ΔU for all phases of HZO for strain in xz- and yz-plain respectively. (e) and (f) show the calculated ΔU for all phases of
ZrO2 for strain in xz- and yz-plain respectively. Areas of zero stress calculations are indicated by (+) and compared with experimental
data (o) from TABLE II, III, and IV. Our choice for the spatial orientation of the m-phase (m) displayed as a thin, black line. The
spatial orientation of the m-phase chosen by Luo et al. 5 and others 6 (m') is displayed by a thick black line. For comparison (b), (d)
and (f) contain the third possible orientation m” which stabilizes the m-phase for all possible strain conditions.
FIG. S3. Calculated Helmholtz free energy difference ΔF at 300 K for all phases and orientations of (a) HfO2, (b) HZO, and (c) ZrO2
under plane strain conditions
E. Surface Energy
FIG. S4. Helmholtz free energy difference ΔF including the surface energy contribution γΩ for all phases as a function of grain
surface area according to our model. The gray vertical lines mark the height h of cylindrical grains with radius r=h/2.
F. Electrical Properties
FIG. S5. E-field dependent Polarization of the m-, t-, and f-phase
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