The effect of strain and oxygen deficiency on the Raman spectrum of monoclinic HfO2 is investigated theoretically using first-principles calculations. 1% in-plane compressive strain applied to a and c axes is found to blue shift the phonon frequencies, while 1% tensile strain does the opposite. The simulations are compared, and good agreement is found with the experimental results of Raman frequencies greater than 110 cm−1 for 50 nm HfO2 thin films. Several Raman modes measured below 110 cm−1 and previously assigned to HfO2 are found to be rotational modes of gases present in air ambient (nitrogen and oxygen). However, localized vibrational modes introduced by threefold-coordinated oxygen (O3) vacancies are identified at 96.4 cm−1 computationally. These results are important for a deeper understanding of vibrational modes in HfO2, which has technological applications in transistors and particularly in resistive random-access memory whose operation relies on oxygen-deficient HfOx.

Hafnium dioxide (HfO2) or hafnia is an important transition metal oxide that finds applications in information technology as a gate dielectric in field effect transistors,1–3 as a resistive switching material in memory devices,4 and in optical coatings.5 As such, it has been studied extensively both theoretically and experimentally.6–19 In particular, fundamental Raman frequencies of the HfO2 monoclinic phase have been previously reported.7,9,10,13,16–20 However, in electronic applications, thin films of HfO2 are typically sputtered or grown by atomic layer deposition (ALD) and are often subject to stoichiometric deviations and strain.6,21–23 Strain, induced by either thermal or lattice mismatch, may shift or even split Raman peaks.24–27 In bulk ceria, for example, lattice expansion causes a substantial softening of the B1u mode.28 In Li2O, when the volume is increased around the superionic transition point, the zone boundary transverse acoustic mode along [110] at the X point is found to be softened.29,30 On the other hand, oxygen vacancies, which are common defects,31,32 can also lead to shifts in phonon frequencies or even to the appearance of normally symmetry-forbidden first-order Raman peaks.9,33,34 In addition, Raman spectroscopy has been previously used to characterize impurities and phase in hafnia films under different annealing conditions.35–38 Thus, it is of fundamental interest to investigate the effects of strain and oxygen vacancies on the phonon spectrum of hafnia. Furthermore, low wavenumber peaks in the range below 110 cm−1 were previously reported experimentally17–19 but were not found computationally in stoichiometric unstrained monoclinic HfO2.7,9–11

In this paper, we carry out first principles calculations and analysis of Raman spectra of hafnia under compressive and tensile strain and in the presence of oxygen vacancies. In the case of strain, we compare the results of calculations with experimental data obtained for HfO2 films deposited by sputtering on thin (50 nm) Au and Pt films on sapphire substrates. The low wavenumber peaks (below 110 cm−1) are carefully examined. This paper is organized as follows: The calculation and experimental methods are described in Sec. II. The phonon spectrum of bulk monoclinic hafnia is discussed in Sec. III. The influence of strain on the zone-center frequencies is discussed in Sec. IV. We discuss the effect of oxygen vacancies in Sec. V and summarize our findings in Sec. VI.

All calculations are carried out using the Vienna ab initio simulation package (VASP).39,40 The exchange-correlation functional is approximated within the local density approximation (LDA), and projector-augmented-wave (PAW) pseudopotentials are used. The electronic configuration for Hf is 5d26s2 and for O is 2s22p4. The use of PAW pseudopotentials allows one to achieve a total energy convergence of 108eV/atom with a cutoff energy of 600 eV. An 8×8×8 Monkhorst k-point mesh is used for the Brillouin zone (BZ) integration for the monoclinic simulation cell. Full structural relaxation is performed until the Hellman-Feynman forces are less than 10−4 eV/Å.

We calculate the phonon dispersion within the harmonic approximation by using the “frozen phonon method.” Force constants in a 2×2×2 supercell are calculated with the Phonopy code.41 The dynamical matrix is computed via the lattice Fourier transformation and frequencies across the BZ are obtained from the eigenvalues of the dynamical matrix.

A neutral oxygen vacancy is simulated by removing one oxygen atom from the supercell, and in our case, this corresponds to a vacancy concentration of 1.56%. Owing to the large size of the supercell, for the defect case, only the Γ-point frequencies are computed as this is sufficient to account for the Raman-active modes.

Because the measured Raman signal of thin HfO2 films is very weak and often only the Raman spectrum of the substrate is detected, we used here a thin metal film to block the substrate signal. In addition, this structure is technologically relevant for resistive random-access memory (RRAM) devices, where thin HfO2 is deposited on a metal electrode. HfO2 films were deposited by room temperature reactive sputtering from an Hf target in an Ar:O2 (7:3) plasma at 4 mT, with a forward RF power of 150 W, on thin (50 nm) Au and Pt films on sapphire substrates. The as-deposited 50 nm thick amorphous HfO2 film did not show any clear Raman features. After annealing at 600 °C for 30 min in air, the HfO2 partially crystallized to the monoclinic phase, which is confirmed by X-ray diffraction (XRD), and Raman features were observed. Measurements were obtained using a Horiba LabRAM HR with 532 nm and 633 nm lasers operated at 2.5 mW and 2 mW, respectively, at room temperature. Structural characterization was carried out using a PANalytical X'Pert PRO X-ray diffractometer equipped with a copper Kα source. The XRD measurement was carried out at a small grazing incidence (1°) to increase the path length of the probing X-ray beam, as well as to minimize diffraction from the underlying substrate.

Bulk crystalline hafnia has three phases: the cubic structure at high temperature (space group Fm3m), the tetragonal structure (space group P42/nmc), and the monoclinic structure at low temperature (space group P21/c). In this work, we only consider the monoclinic structure. We list the optimized structural parameters of hafnia compared with experimental data in Table I. The Hf cation is at the Wyckoff position 4e(0.277,0.042,0.208), while the threefold-coordinated O3 anion and fourfold-coordinated O4 anion are at 4e(0.073,0.341,0.337) and 4e(0.446,0.759,0.483), respectively. This agrees well with the data reported previously.7 The theoretical volume is a little smaller than the experimental value, as the LDA tends to overbind, resulting in a smaller lattice constant. The calculation corresponds to zero temperature while experiments are typically done at room temperature. Note, however, that the generalized gradient approximation (GGA) often matches better with experimental lattice constants than LDA, but LDA gives a better description of phonons in hafnia.7 

TABLE I.

Structural parameters of monoclinic HfO2. Th. = theoretical and Exp. = experimental.

a (Å)b (Å)c (Å)β (deg)From
5.029 5.132 5.183 99.48 Th.10  
5.106 5.165 5.281 99.35 Th.7  
5.116 5.172 5.295 99.18 Exp.51  
5.119 5.169 5.29 99.25 Exp.52  
5.117 5.172 5.284 99.37 Exp.53  
5.025 5.118 5.192 99.52 Present theory 
a (Å)b (Å)c (Å)β (deg)From
5.029 5.132 5.183 99.48 Th.10  
5.106 5.165 5.281 99.35 Th.7  
5.116 5.172 5.295 99.18 Exp.51  
5.119 5.169 5.29 99.25 Exp.52  
5.117 5.172 5.284 99.37 Exp.53  
5.025 5.118 5.192 99.52 Present theory 

In Fig. 1(a), we show the phonon spectrum of monoclinic hafnia computed including only the short range contribution to the dynamical matrix (blue line). The BZ path of the calculation starts and ends at the Γ(0, 0, 0) point, going through B(0, 0, 0.5), A(0.5, 0, 0.5), E(0.5, 0.5, 0.5), and Y(0.5, 0, 0) high symmetry points of the BZ. From group theoretical analysis, the zone-center modes can be decomposed as

Γ=9Ag+9BgRaman+8Au+7BuIR+Au+2BuAcoustic.

The phonon density of states is plotted in Fig. 1(b). We can see a quasi-gap around 350 cm1 which divides the spectrum into low frequency part and high frequency part. As pointed out in Ref. 10, low frequency modes involve predominantly metal atoms while high frequency modes are associated with movement of oxygen atoms. Among these low frequency modes, however, some are still anomalous, as along with hafnium, oxygen ions also exhibit large displacements. We illustrate the eigenvector corresponding to the lowest frequency Raman-active mode at 127 cm−1 in Fig. 2. In this mode, the threefold-coordinated oxygen atoms (marked red) show the most significant displacement, while displacements of hafnium atoms are only half of that. The fourfold-coordinated oxygen atoms (magenta colored) practically do not move.

FIG. 1.

(a) Calculated phonon dispersion of monoclinic hafnia. Blue, black, and red lines represent no strain, 1% compressive, and tensile strain applied to a and c axes case, respectively. (b) Phonon density of states for no strain case.

FIG. 1.

(a) Calculated phonon dispersion of monoclinic hafnia. Blue, black, and red lines represent no strain, 1% compressive, and tensile strain applied to a and c axes case, respectively. (b) Phonon density of states for no strain case.

Close modal
FIG. 2.

Relative atomic displacements of the lowest Raman mode. Arrows denote the direction of the movement at each atom and the length is proportional to displacement amplitude. The monoclinic angle is between a and c axes. Bonds are also plotted to show different coordinations of the O atom. 3-folded O atoms are marked by red and 4-folded O atoms are marked by magenta.

FIG. 2.

Relative atomic displacements of the lowest Raman mode. Arrows denote the direction of the movement at each atom and the length is proportional to displacement amplitude. The monoclinic angle is between a and c axes. Bonds are also plotted to show different coordinations of the O atom. 3-folded O atoms are marked by red and 4-folded O atoms are marked by magenta.

Close modal

In Table II, we list the frequencies of the Raman-active modes obtained in our calculation along with two sets of previous experimental results. Agreement between the calculation and experiments is very good.

TABLE II.

Calculated Raman frequencies (in cm−1) for monoclinic HfO2 and available experimental data. Th. = theoretical and Exp. = experimental.

AgTh.10 Th.7 Exp.16 Exp.13 Present workBgTh.10 Th.7 Exp.16 Exp.13 Present theory
133 128 112 113 127 136 131 133 133 135 
140 142 135 133 138 171 175 167 164 169 
153 152 150 149 153 246 250 243 242 250 
257 261 257 256 260 339 380 335 336 339 
361 326 322 323 352 421 424 397 398 414 
410 423 381 382 397 537 533 522 520 516 
512 514 498 498 498 578 570 550 551 564 
601 608 581 577 581 663 667 642 640 642 
696 738 674 672 663 785 821 776 872 780 
AgTh.10 Th.7 Exp.16 Exp.13 Present workBgTh.10 Th.7 Exp.16 Exp.13 Present theory
133 128 112 113 127 136 131 133 133 135 
140 142 135 133 138 171 175 167 164 169 
153 152 150 149 153 246 250 243 242 250 
257 261 257 256 260 339 380 335 336 339 
361 326 322 323 352 421 424 397 398 414 
410 423 381 382 397 537 533 522 520 516 
512 514 498 498 498 578 570 550 551 564 
601 608 581 577 581 663 667 642 640 642 
696 738 674 672 663 785 821 776 872 780 

In the monoclinic phase of hafnia, the three crystallographic axes are not orthogonal to each other. Here, we model the case when the out-of-plane direction of the film is along the b axis, and in-plane biaxial strain is applied to a and c axes. Strain is fixed to be 1%, and both the tensile and compressive strains are considered. The lattice constants in the ac plane are changed by 1% of the original value, while the monoclinic angle is fixed. We optimize the b lattice constant, letting the ions move freely until the ionic forces are less than 10−4 eV/Å. The corresponding change in the b lattice constant is 0.75 ± 0.01%, which means that the Poisson ratio is 0.75. From the elastic energy theory, with a biaxial strain applied to a and c axes, the Poisson ratio ν=εbεa could be expressed as C12+C23C22, where C stands for the elastic constants. From our calculation, C12, C23, and C22 are 489 GPa, 180 GPa, and 201 GPa, respectively. Following this, the Poisson ratio ν is calculated as 0.78, very close to what we get from relaxation. For comparison, by taking elastic constants from Refs. 42 and 43, the Poisson ratio ν is calculated to be 0.82 and 0.88, respectively. The phonon spectrum is then calculated as previously described.

The effect of strain on the phonon spectrum is shown in Fig. 1(a). The average of frequencies of three acoustic modes at the Γ point is 0.06 cm−1; we use this as an error estimate of the calculation. The main conclusion is that 1% strain applied to a and c axes does not significantly change the phonon spectrum. Compressive strain shifts all modes up in frequency, while tensile strain does exactly the opposite. This is an expected result considering the overall shape of interatomic potentials. We list the frequencies of the main Raman active modes in Table III.

TABLE III.

Calculated Raman frequencies of monoclinic hafnia (in cm−1) in four different cases: without strain, with 1% in-plane biaxial compressive and tensile strain applied to a and c axes, and strain applied on sample (sample strain). Calculation results are compared with eleven identified Raman peaks in the experiment.

ModeNo strainCompressive strainTensile strainSample strainMeasured% Error
Ag127 137 111 137   
Ag138 147 136 144   
Ag153 162 148 158 150 5.6 
Ag260 268 257 267 258 3.4 
Ag352 364 337 363   
Ag397 405 388 405 383 5.4 
Ag498 499 495 500 498 0.4 
Ag581 580 582 588 578 1.9 
Ag663 666 659 668 667 0.1 
Bg135 138 132 139 135 2.9 
Bg169 172 167 171   
Bg250 251 248 251 240 4.4 
Bg339 343 333 344 326 5.2 
Bg414 419 402 419 399 4.8 
Bg516 514 509 515   
Bg564 574 553 575   
Bg642 652 634 655 638 2.6 
Bg780 778 779 783   
ModeNo strainCompressive strainTensile strainSample strainMeasured% Error
Ag127 137 111 137   
Ag138 147 136 144   
Ag153 162 148 158 150 5.6 
Ag260 268 257 267 258 3.4 
Ag352 364 337 363   
Ag397 405 388 405 383 5.4 
Ag498 499 495 500 498 0.4 
Ag581 580 582 588 578 1.9 
Ag663 666 659 668 667 0.1 
Bg135 138 132 139 135 2.9 
Bg169 172 167 171   
Bg250 251 248 251 240 4.4 
Bg339 343 333 344 326 5.2 
Bg414 419 402 419 399 4.8 
Bg516 514 509 515   
Bg564 574 553 575   
Bg642 652 634 655 638 2.6 
Bg780 778 779 783   

Experimentally, the film is crystallized at 600 °C first and then cooled down to room temperature. XRD determines the HfO2 thin film strain based on the known crystal structure. Since the metallic substrate diffracts at larger 2θ, we focus on the diffraction peaks at lower angles, which can be unambiguously attributed to HfO2. The multiple peaks observed from the out-of-plane XRD data (Fig. 3) correspond well to the known monoclinic phase of HfO2. No texture is observed as the relative intensities of the peaks observed in the measurement correspond well to the powder diffraction,44 demonstrating that the film is polycrystalline in nature. The calculated lattice constants a, b, and c for HfO2 deposited on Pt are measured to be 5.1 ± 0.1 Å, 5.19 ± 0.06 Å, and 5.22 ± 0.05 Å, respectively, with a tilt angle β of 99.0°± 0.2°. Compared with the experimental bulk values in Table I, a and c axes are under compressive strain while the b axis is strained slightly. This is not unexpected as the material is deposited amorphous and becomes polycrystalline upon annealing.

FIG. 3.

X-ray diffraction scan of sputtered hafnia on a platinum substrate, measured at a grazing incidence (1°). The scan demonstrates that polycrystalline monoclinic hafnia is formed after annealing.

FIG. 3.

X-ray diffraction scan of sputtered hafnia on a platinum substrate, measured at a grazing incidence (1°). The scan demonstrates that polycrystalline monoclinic hafnia is formed after annealing.

Close modal

Figure 4(a) displays the Raman spectra of the HfO2 deposited on Pt, measured using a 532 nm laser. The Lorentzian fit to the HfO2 on the Pt spectrum is also shown in Fig. 4(a), offset vertically for clarity. Measurements using the 633 nm laser show the same spectra. In order to have a fair comparison, we strain the simulation cell the same way as is measured experimentally and compute Raman frequencies, which are listed in Table III. The result is very similar to that obtained for the compressively strained material. This is expected, as both a and c axes are under compressive strain. Eleven out of the eighteen calculated Raman modes are identified in the measured spectrum. Fitted peak positions are summarized in Table III and compared against the calculated peaks for the case of a cell having the strain measured experimentally. We note that calculations were carried out at zero temperature, whereas the measurement was done at room temperature. No measureable peak shifts were observed following an additional annealing step in Ar ambient at 600 °C for 2 h, suggesting that the main Raman modes in HfO2 are insensitive to minor changes in the film stoichiometry, unlike TaOx45 and SrTiO3.33 Previous studies have shown that annealing SrTiO3 in reducing ambient results in the appearance of forbidden first order Raman peaks and features associated with oxygen vacancies.33 Similarly, the Raman spectra of sputtered TaOx films show a clear dependence on x by varying the oxygen pressure during the sputtering process.45 

FIG. 4.

Experimental Raman spectra of HfO2. (a) Bottom: measured Raman spectra of 50 nm HfO2 on Pt using 532 nm laser (blue) and top: Lorentzian fit (green—for each peak, black—sum over all Lorentzians). Spectra are vertically offset for clarity. The HfO2 was heated to 600 °C and then cooled to room temperature, forming a polycrystalline monoclinic phase. (See Fig. 3.) (b) Low wavenumber range: measured HfO2 [blue, the same as in (a)] vs. control experiment without sample and without objective in the laser path (red) showing similar features below 120 cm−1. (c) Control measurement of the low wavenumber Raman spectrum [the same as the red curve in (b), measured data in black solid line] compared with rotational Raman modes of N2 (red dashed) and O2 (blue dashed-dotted). The low wavenumber modes at 59, 83, and 107 cm−1 are clearly observed in the Raman spectra of the thin HfO2 sample [marked “ambient peaks” in (a)] and are of comparable intensity due to the weak Raman response of the HfO2 and the enhanced Raman signal of the rotational modes, where N2 and O2 modes overlap.

FIG. 4.

Experimental Raman spectra of HfO2. (a) Bottom: measured Raman spectra of 50 nm HfO2 on Pt using 532 nm laser (blue) and top: Lorentzian fit (green—for each peak, black—sum over all Lorentzians). Spectra are vertically offset for clarity. The HfO2 was heated to 600 °C and then cooled to room temperature, forming a polycrystalline monoclinic phase. (See Fig. 3.) (b) Low wavenumber range: measured HfO2 [blue, the same as in (a)] vs. control experiment without sample and without objective in the laser path (red) showing similar features below 120 cm−1. (c) Control measurement of the low wavenumber Raman spectrum [the same as the red curve in (b), measured data in black solid line] compared with rotational Raman modes of N2 (red dashed) and O2 (blue dashed-dotted). The low wavenumber modes at 59, 83, and 107 cm−1 are clearly observed in the Raman spectra of the thin HfO2 sample [marked “ambient peaks” in (a)] and are of comparable intensity due to the weak Raman response of the HfO2 and the enhanced Raman signal of the rotational modes, where N2 and O2 modes overlap.

Close modal

Interestingly, some features appear in the measured Raman spectra of our HfO2 films below 110 cm−1. In particular, Raman modes at ∼59 cm−1, ∼83 cm−1, and ∼107 cm−1 consistently appear across different measurements on different substrates. The ∼82 cm−1 Raman line was also reported by Quintard et al.17 who interpreted this line as fundamental, even though bulk unstrained HfO2 does not have any modes at that low a wavenumber. A low wavenumber peak in the range of 105–108 cm−1 was also reported by Refs. 1719.

According to our theoretical results, the strain found in our sample is not capable of causing a frequency downshift of that magnitude. We further investigated experimentally the weak Raman lines below 110 cm−1 as shown in Figs. 4(b) and 4(c). We carried out a control experiment at high laser power (20 mW) and long accumulation (15 min) without sample and objective in the laser path. We found the same low wavenumber peaks, namely, at ∼59, 83, and 107 cm−1 and few other smaller features [Fig. 4(b)]. We attribute these low wavenumber modes to rotational Raman modes of the oxygen and nitrogen molecules present in air ambient as discussed below. We note that the presence of these “ambient peaks” does not exclude the possibility of having fundamental HfO2 modes in that range (as reported in Refs. 17 and 18) however measurements carried out in air ambient cannot detect such modes.

Figure 4(c) shows very good agreement between the low wavenumber features found in our control measurement and reported rotational Raman modes of N2 and O2 from the literature.46,47 In particular, the modes at ∼59 cm−1, ∼83 cm−1, and ∼107 cm−1 are enhanced due to the overlap between the O2 and N2 modes. It is evident from Fig. 4(a) that the same peaks (marked as “ambient peaks”) are of comparable intensity to the fundamental HfO2 peaks. This is an important note to any Raman measurement carried out in air of a material having low absorption and crystallinity, such that the Raman signal of the ambient may become comparable to the signal of the sample. Here, the periodic features at low wavenumbers (frequency spacing Δω  ≈  24 cm−1 for the 3 peaks outlined above) are a hallmark of rotational Raman modes. The vibrational Raman modes of these gases present in air are located at much higher wavenumbers, but their rotational modes are spectrally located at the low wavenumber range measured here. Figure 4(c) shows that the specific modes which are comparable to the HfO2 signal correspond to the overlap between N2 and O2 modes, where the N2 modes have a periodicity of ∼8 cm−1 and the O2 have a periodicity of ∼12 cm−1, resulting in the enhanced Raman features with a periodicity of ∼24 cm−1.

Thin HfO2 films are often non-stoichiometric, and the most prevalent defect is oxygen vacancies. In monoclinic hafnia, there are two non-equivalent oxygen sites: (i) threefold-coordinated oxygen O3, bonded with the nearest-neighbor Hf atoms in an almost planar configuration; and (ii) fourfold-coordinated oxygen O4, bonded with its Hf neighbors in a distorted tetrahedral configuration. This means that one can have two different types of oxygen vacancies.48,49 Also, since the formation energies for neutral O3 and O4 vacancies are very close (9.36 eV and 9.34 eV, respectively48), it is worth investigating the effect of both vacancy types on the phonon spectrum. For ease of comparison, we also compute zone-center frequencies in pure hafnia, in the same 2×2×2 supercell. For pure hafnia, there are 288 modes in a 96 atom cell. However, all Γ-point frequencies are related to those computed for the primitive cell by a corresponding reciprocal lattice vector. As we double the lattice parameters, the new BZ becomes twice as small and the high symmetry points of the primitive are translated back to the Γ-point [Fig. 5(a)]. Therefore, the zone-center frequencies of this calculation are the frequencies from the eight high symmetry points of the primitive cell: (0, 0, 0), (0.5, 0, 0), (0, 0.5, 0), (0, 0, 0.5), (0.5, 0.5, 0), (0, 0.5, 0.5), (0.5, 0, 0.5), and (0.5, 0.5, 0.5). We illustrate how the hafnia phonon spectrum “folds” along the direction from the high symmetry point Γ to E in Fig. 5(b) (up to 200 cm−1). Blue and red lines represent the phonon spectrum in the primitive cell and in the supercell, respectively. The green arrows show how the original spectrum is translated back to the new first BZ (FBZ).

FIG. 5.

(a) The first Brillouin zone (FBZ) of the conventional monoclinic (MCL) lattice. Under this convention, (0.5, 0, 0), (0, 0.5, 0), (0, 0, 0.5), (0.5, 0.5, 0), (0, 0.5, 0.5), (0.5, 0, 0.5), and (0.5, 0.5, 0.5) are the Z, X, Y, A, C, D, and E points in the figure. Black lines represent the FBZ of the primitive cell, while the red lines denote the FBZ of the doubled supercell. The primitive cell and supercell reciprocal lattice vectors bi and bi are also shown with black and red arrows, respectively, and bi=12bi. We can see that the high symmetry points in the primitive FBZ can be translated back to the Gamma point by purple vectors, which is the combination of bi. (b) A schematic plot about the phonon spectrum folding along the direction from E¯ (−0.5, −0.5, −0.5) to Γ to E (0.5, 0.5, 0.5). Green arrow represents the reciprocal lattice vector G, indicating how the spectrum “folds” from the large Brillouin zone into the small Brillouin zone.

FIG. 5.

(a) The first Brillouin zone (FBZ) of the conventional monoclinic (MCL) lattice. Under this convention, (0.5, 0, 0), (0, 0.5, 0), (0, 0, 0.5), (0.5, 0.5, 0), (0, 0.5, 0.5), (0.5, 0, 0.5), and (0.5, 0.5, 0.5) are the Z, X, Y, A, C, D, and E points in the figure. Black lines represent the FBZ of the primitive cell, while the red lines denote the FBZ of the doubled supercell. The primitive cell and supercell reciprocal lattice vectors bi and bi are also shown with black and red arrows, respectively, and bi=12bi. We can see that the high symmetry points in the primitive FBZ can be translated back to the Gamma point by purple vectors, which is the combination of bi. (b) A schematic plot about the phonon spectrum folding along the direction from E¯ (−0.5, −0.5, −0.5) to Γ to E (0.5, 0.5, 0.5). Green arrow represents the reciprocal lattice vector G, indicating how the spectrum “folds” from the large Brillouin zone into the small Brillouin zone.

Close modal

To establish whether or not a mode is introduced by a vacancy, we compare frequencies computed for both the vacancy-containing and defect-free cells. For a specific frequency in the spectrum of a cell with a vacancy, if the corresponding mode with a similar frequency (± 2 cm−1) can be found in the vacancy-free spectrum, then this mode is thought to be a bulk mode and the small shift in frequency is the result of perturbation introduced by the vacancy. However, if no related bulk modes are found and there is a large gap (10 cm−1 or larger) between the frequencies of two cells, then the mode is directly related to the vacancy. We focus on the low frequency modes first (which are metal-dominated). All frequencies below 127 cm−1 (the first Raman mode of the primitive cell) for O3 and O4 vacancy cells are listed in Table IV. The defect-free calculation results are also included for comparison. We can see that in most cases, the differences between the frequencies calculated for these cells are very small (± 2 cm−1). The majority of the modes are only slightly perturbed by the defect. However, for the cell with an O3 vacancy, a very low frequency mode (see Table V) is found at 96.4 cm−1. It is clearly distinct from the nearby bulk modes at 93.8 cm−1 and 103.2 cm−1. Note that only the short range effects are captured, as we do not include the long range correction to the dynamical matrix. However, this is not very rigorous, and to obtain the quantitative description, we use the following approach.

TABLE IV.

Zone-centered frequencies (in cm−1) below 127 cm−1 for the pure hafnia supercell case and O3 and O4 vacancy cases. The corresponding k-points in the primitive cell are also listed.

k-points in primitive cellDefect-free supercell caseO3 oxygen vacancyO4 oxygen vacancyk-points in primitive cellDefect-free supercell caseO3 oxygen vacancyO4 oxygen vacancy
(0.5,0,0) 83.0 81.3 82.6 (0.5,0,0.5) 107.1 106.3 106.3 
(0,0,0.5) 84.5 83.2 82.9 (0.5,0,0.5) 107.1 108.2 106.8 
(0,0,0.5) 84.5 83.9 84.1 (0,0.5,0) 111.2 110.4 109.2 
(0.5,0.5,0.5) 92.3 90.2 90.2 (0,0.5,0) 111.2 112.4 110.9 
(0.5,0.5,0.5) 92.3 91.5 90.6 (0.5,0,0) 112.9 112.7 111.3 
(0.5,0.5,0) 93.5 92.4 92.3 (0.5,0.5,0) 113.8 113.3 112.6 
(0.5,0.5,0) 93.5 92.8 92.9 (0.5,0.5,0) 113.8 114.3 113.1 
(0,0.5,0) 93.8 93.1 93.3 (0.5,0,0) 114.7 114.8 114.0 
(0,0.5,0) 93.8 93.6 93.8 (0.5,0.5,0.5) 115.8 115.3 115.1 
(0,0,0.5) 103.2 101.2 101.7 (0.5,0.5,0.5) 115.8 117.1 115.7 
(0,0,0.5) 103.2 102.9 103.1 (0.5,0.5,0.5) 120.8 119.7 119.3 
(0.5,0,0) 105.7 105.2 104.3 (0.5,0.5,0.5) 120.8 120.9 120.5 
(0,0.5,0.5) 106.3 105.9 105.0 (0.5,0,0.5) 122.5 121.5 121.1 
(0,0.5,0.5) 106.3 106.3 105.4 (0.5,0,0.5) 122.5 123.5 122.0 
(0.5,0,0) 83.0 81.3 82.6 (0.5,0,0.5) 107.1 106.3 106.3 
k-points in primitive cellDefect-free supercell caseO3 oxygen vacancyO4 oxygen vacancyk-points in primitive cellDefect-free supercell caseO3 oxygen vacancyO4 oxygen vacancy
(0.5,0,0) 83.0 81.3 82.6 (0.5,0,0.5) 107.1 106.3 106.3 
(0,0,0.5) 84.5 83.2 82.9 (0.5,0,0.5) 107.1 108.2 106.8 
(0,0,0.5) 84.5 83.9 84.1 (0,0.5,0) 111.2 110.4 109.2 
(0.5,0.5,0.5) 92.3 90.2 90.2 (0,0.5,0) 111.2 112.4 110.9 
(0.5,0.5,0.5) 92.3 91.5 90.6 (0.5,0,0) 112.9 112.7 111.3 
(0.5,0.5,0) 93.5 92.4 92.3 (0.5,0.5,0) 113.8 113.3 112.6 
(0.5,0.5,0) 93.5 92.8 92.9 (0.5,0.5,0) 113.8 114.3 113.1 
(0,0.5,0) 93.8 93.1 93.3 (0.5,0,0) 114.7 114.8 114.0 
(0,0.5,0) 93.8 93.6 93.8 (0.5,0.5,0.5) 115.8 115.3 115.1 
(0,0,0.5) 103.2 101.2 101.7 (0.5,0.5,0.5) 115.8 117.1 115.7 
(0,0,0.5) 103.2 102.9 103.1 (0.5,0.5,0.5) 120.8 119.7 119.3 
(0.5,0,0) 105.7 105.2 104.3 (0.5,0.5,0.5) 120.8 120.9 120.5 
(0,0.5,0.5) 106.3 105.9 105.0 (0.5,0,0.5) 122.5 121.5 121.1 
(0,0.5,0.5) 106.3 106.3 105.4 (0.5,0,0.5) 122.5 123.5 122.0 
(0.5,0,0) 83.0 81.3 82.6 (0.5,0,0.5) 107.1 106.3 106.3 
TABLE V.

Frequencies (in cm−1) of vacancy-related modes in the O3 vacancy cell and corresponding inverse participation ratio (IPR) values. Modes with large IPR values are highlighted in bold.

FrequencyIPRFrequencyIPR
96.4 0.057 502.3 0.03 
209.3 0.043 504.8 0.031 
211.2 0.032 552.8 0.025 
213.7 0.033 568.5 0.027 
233.3 0.047 627.9 0.027 
234.4 0.038 680.4 0.096 
281.4 0.041 721.8 0.03 
315.9 0.041 734.9 0.033 
345.6 0.042 739.5 0.038 
411.6 0.051 751.2 0.069 
412.7 0.032 756.6 0.049 
449.4 0.04 788.2 0.076 
485.4 0.031 804.2 0.033 
493.6 0.03   
FrequencyIPRFrequencyIPR
96.4 0.057 502.3 0.03 
209.3 0.043 504.8 0.031 
211.2 0.032 552.8 0.025 
213.7 0.033 568.5 0.027 
233.3 0.047 627.9 0.027 
234.4 0.038 680.4 0.096 
281.4 0.041 721.8 0.03 
315.9 0.041 734.9 0.033 
345.6 0.042 739.5 0.038 
411.6 0.051 751.2 0.069 
412.7 0.032 756.6 0.049 
449.4 0.04 788.2 0.076 
485.4 0.031 804.2 0.033 
493.6 0.03   

A more accurate way to identify the vacancy-related modes is by calculating the inner product between the eigenvectors of a vacancy-containing cell with those of a defect-free cell. We use Φα(q), where α=1,2,,288, to represent 288 phonon modes at the Γ point of a 96-atom, defect-free cell. Every eigenvector has 288 components uαi(q) corresponding to 3 × N degrees of freedom. Similarly, in a vacancy cell, 285 phonon modes at the Γ point are recorded as Φα(q), where α=1,2,,285 and they have 285 components uαi(q). Since the eigenvector of a vacancy cell has 3 fewer components, we set them to zero. We calculate the inner product between Φα(q) and Φβ(q) and define a distribution function fαβ=Φα(q)|Φβ(q)2 for Φα(q). If the distribution is close to a delta function δαβ, it indicates that Φα(q) is similar to a bulk-like mode. If, on the other hand, the distribution is broad, it manifests that the mode could not be simply represented by a single, bulk-like mode but is related to a vacancy. Overall β=1288fαβ=1. In Figs. 6(a) and 6(b), we use 3D plots to show fαβ for O3 and O4 vacancy cells, respectively. We find that for the O4-based vacancy case, most of the vacancy-related modes appear in the high frequency region of the spectrum, while for the O3-based vacancy case, the modes are distributed more evenly. We calculate gα=maxβ(fαβ1) for all Φα(q) in both O3 and O4 vacancy cells as shown in Figs. 6(c) and 6(d). A large gα corresponds to a broad distribution, while a small gα(1) corresponds to a narrow distribution. We use 4 as a cutoff value for gα. Combining these two methods, all vacancy-related frequencies are listed in Tables V and VI for O3 and O4 vacancies, respectively.

FIG. 6.

(a) Distribution of square of the inner product fαβ in the O3 vacancy cell. (b) Distribution of square of the inner product fαβ in the O4 vacancy cell. (c) gα for vacancy-related modes in the O3 vacancy cell. (d) gα for vacancy-related modes in the O4 vacancy cell.

FIG. 6.

(a) Distribution of square of the inner product fαβ in the O3 vacancy cell. (b) Distribution of square of the inner product fαβ in the O4 vacancy cell. (c) gα for vacancy-related modes in the O3 vacancy cell. (d) gα for vacancy-related modes in the O4 vacancy cell.

Close modal
TABLE VI.

Frequencies (in cm−1) of vacancy-related modes in the O4 vacancy cell and corresponding inverse participation ratio (IPR) values. Modes with large IPR values are highlighted in bold.

FrequencyIPRFrequencyIPR
178.2 0.038 647.2 0.027 
377.6 0.042 647.8 0.024 
417.3 0.032 684.6 0.022 
476.3 0.06 706.4 0.036 
499.2 0.029 731.3 0.031 
519.8 0.031 733.0 0.036 
533.0 0.028 737.0 0.043 
542.4 0.032 755.6 0.051 
553.9 0.029 758.8 0.044 
634.6 0.033 790.2 0.045 
FrequencyIPRFrequencyIPR
178.2 0.038 647.2 0.027 
377.6 0.042 647.8 0.024 
417.3 0.032 684.6 0.022 
476.3 0.06 706.4 0.036 
499.2 0.029 731.3 0.031 
519.8 0.031 733.0 0.036 
533.0 0.028 737.0 0.043 
542.4 0.032 755.6 0.051 
553.9 0.029 758.8 0.044 
634.6 0.033 790.2 0.045 

To characterize the degree of localization of the vacancy-related modes quantitatively, we calculate the inverse participation ratio (IPR) for all the modes listed in Tables V and VI. The IPR is defined as Pk−1=N(α=13uiα,k2)2 (note that the eigenvector is normalized), where N is the total number of atoms and α refers to three spatial directions.50 For the phonon mode where all atoms contribute equally (have the same amplitude), IPR is 1/N, and in our case, it is around 0.01. If there are only m atoms that vibrate in the mode, then IPR is 1/m. For comparison, we compute IPR values for all Γ-point modes in a defect-free cell. For most of them, the IPR is less than 0.03 and the maximum value is less than 0.05. We use this value as a cutoff for identifying localized modes. From Tables V and VI, we find that modes at 96.4 cm−1, 411.6 cm−1, 680.4 cm−1, 751.2 cm−1, and 788.2 cm−1 in the O3 vacancy cell and 476.3 cm−1, 755.6 cm−1 in the O4 vacancy cell have large IPR values, which indicates that a significant vibration is only around a few atoms for these modes. We carefully checked atomic movements of each vacancy-related, high-IPR-value mode and concluded that the 96.4 cm−1 mode in the O3 vacancy cell and 755.6 cm−1 mode in the O4 vacancy cell are localized defect modes, with atomic displacements localized around the defect site. In Figs. 7(a) and 7(b), using bar plots we show relative displacement amplitudes in real space for each atom in these two modes (1miui,x2+ui,y2+ui,z2, where mi is the atomic mass and ui is the entry in the eigenvector corresponding to the ith atom of the basis, with the eigenvector normalized to unity). For the 96.4 cm−1 mode, one Hf atom and one O atom move more significantly than others, and we mark these two as black in Fig. 7(c). We note that the Hf atom is directly bonded to the missing 3-fold oxygen and the other O atom is also very close to a vacancy (2.8 Å). For the 755.6 cm−1 mode, as this is a high frequency mode, all 5 atoms that move most significantly are oxygens. In Fig. 7(d), except for the blue O atom, which is 4.5 Å from the vacancy site, all of other 4 black O atoms are close to the vacancy site (distances to vacancy are 2.5 Å, 2.9 Å, 2.9 Å, and 3.2 Å).

FIG. 7.

(a) Bar plot of the mode. The X axis represents atomic number in the cell and first 32 atoms are Hf atoms, while other 63 atoms are O atoms. Colors of different atoms are selected according to (c). (b) Bar plot of the mode. Colors of different atoms are selected according to (d). (c) The simulation cell containing an O3 vacancy. Red balls represent oxygen atoms and cyan balls represent Hf atoms. The O3 vacancy is created by removing the grey atom. Atoms that have significant displacements in the 96.4 cm−1 mode are highlighted with black. (d) The simulation cell containing an O4 vacancy. The O4 vacancy is created by removing the grey atom. Atoms that have significant displacements in 755.6 cm−1 mode are highlighted.

FIG. 7.

(a) Bar plot of the mode. The X axis represents atomic number in the cell and first 32 atoms are Hf atoms, while other 63 atoms are O atoms. Colors of different atoms are selected according to (c). (b) Bar plot of the mode. Colors of different atoms are selected according to (d). (c) The simulation cell containing an O3 vacancy. Red balls represent oxygen atoms and cyan balls represent Hf atoms. The O3 vacancy is created by removing the grey atom. Atoms that have significant displacements in the 96.4 cm−1 mode are highlighted with black. (d) The simulation cell containing an O4 vacancy. The O4 vacancy is created by removing the grey atom. Atoms that have significant displacements in 755.6 cm−1 mode are highlighted.

Close modal

Using density functional theory, we investigate the phonon spectra of HfO2 in the presence of oxygen vacancies and strain, comparing them against experimental Raman measurements of thin HfO2 films. The measured Raman spectra show low wavenumber features (below 110 cm−1) which we assign to rotational Raman modes of the gases present in air ambient rather than the HfO2 film. It is found that a 1% in-plane tensile strain to a and c axes can result in down-shift while a 1% compressive strain can cause the up-shift of the Raman active modes. For the threefold-coordinated oxygen vacancy case, a low frequency metal-dominated 96.4 cm−1 mode, which is 30 cm−1 lower than the first Raman active mode of stoichiometric hafnia, is predicted. For the four-coordinated oxygen vacancy case, a high frequency, oxygen-dominated mode at 755.6 cm−1 is predicted. We reveal these two modes to be highly localized by calculating the inverse participation ratio (IPR) and by analyzing atomic displacements in real space, verifying that they could be introduced by oxygen vacancies.

We thank Agham Posadas for critically reading the manuscript and Ajay Sood for helpful discussions. Support for this work was provided through Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research, and Basic Energy Sciences under Award No. DESC0008877. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation (NSF) under Award No. ECCS-1542152. E. Yalon acknowledges the support of Ilan Ramon Fulbright Fellowship and the Andrew and Erna Finci Viterbi Foundation. E. Pop acknowledges support from the NSF DMREF Grant No. 1534279 and the Stanford SystemX Alliance.

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