Crystal structure of monoclinic hafnia (HfO2) revisited with synchrotron X-ray, neutron diffraction and first-principles calculations

Pathak, S.; Das, P.; Das, T.; Mandal, G.; Joseph, B.; Sahu, M.; Kaushik, S.D.; Siruguri, V.

doi:10.1107/S2053229620013960

Crystal structure of monoclinic hafnia (HfO₂) revisited with synchrotron X-ray, neutron diffraction and first-principles calculations

S. Pathak, P. Das, T. Das, G. Mandal, B. Joseph, M. Sahu, S. D. Kaushik and V. Siruguri

A study on the crystal structure of monoclinic HfO₂ has been performed using synchrotron X-ray and neutron diffraction data separately, as well as a combination of both. The precision of the structural parameters increases significantly due to application of the neutron diffraction technique. The experimental oxygen positions in HfO₂, derived precisely, are visualized only by semi-local density functional calculations in terms of the calculated electronic band gap, but are not captured as accurately by using hybrid functionals.

Keywords: hafnia; neutron diffraction; powder diffraction; synchrotron radiation; Rietveld refinement; DFT; hybrid functional; band gap.

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Supporting information

	Crystallographic Information File (CIF) https://doi.org/10.1107/S2053229620013960/ov3141sup1.cif Contains datablocks global, I_synchrotron, I_Neutron, Neutron, I_combined_b, combined
	Structure factor file (CIF format) https://doi.org/10.1107/S2053229620013960/ov3141I_Neutronsup2.hkl Contains datablock I_Neutron
	Structure factor file (CIF format) https://doi.org/10.1107/S2053229620013960/ov3141I_synchrotronsup3.hkl Contains datablock I_synchrotron
	Rietveld powder data file (CIF format) https://doi.org/10.1107/S2053229620013960/ov3141combinedsup4.rtv Contains datablock I
	Portable Document Format (PDF) file https://doi.org/10.1107/S2053229620013960/ov3141sup5.pdf Supplementary material

CCDC reference: 2039450

Computing details top

Program(s) used to solve structure: GSAS (Larson & Von Dreele, 2004) for I_synchrotron, I_Neutron. Program(s) used to refine structure: GSAS (Larson & Von Dreele, 2004) for I_synchrotron, I_Neutron.

Hafnium oxide (I_synchrotron) top

Crystal data top

HfO₂	Z = 4
M_r = 210.49	D_x = 10.129 Mg m⁻³
Monoclinic, P2₁/c	Synchrotron radiation, λ = 0.5007 Å
a = 5.1145 (2) Å	T = 295 K
b = 5.1682 (2) Å	Particle morphology: sphere
c = 5.2900 (2) Å	white
β = 99.211 (3)°	cylinder, 25 × 0.1 mm
V = 138.03 (1) Å³	Specimen preparation: Prepared at 295 K and 101.3 kPa

Data collection top

Measurement at synchrotron beamline diffractometer	Data collection mode: transmission
Radiation source: synchrotron, Xpress beamline, Elettra	Scan method: Stationary detector
Si-crystal monochromator	Absorption correction: for a cylinder mounted on the φ axis GSAS Absorption/surface roughness correction: function number 0 (Lobanov & alte da Veiga, 1998).6th European Powder Diffraction Conference, 22-25 August 1998, Budapest, Hungary, Abstract P12-16.) Debye-Scherrer absorption correction Term (= MU.r/wave) = 0.20573E-01 Correction is not refined.
Specimen mounting: Capillary	T_min = 0.983, T_max = 0.983

Refinement top

Least-squares matrix: full	Profile function: CW Profile function number 2 Profile coefficients for Simpson's rule integration of pseudovoigt function Howard, C. J. (1982). J. Appl. Cryst., 15, 615–620. Thompson, P., Cox, D. E. & Hastings, J. B. (1987). J. Appl. Cryst., 20, 79–83. Peak tails are ignored where the intensity is below 0.0010 times the peak.
R_p = 0.081	41 parameters
R_wp = 0.111	0 restraints
R_exp = 0.091	(Δ/σ)_max < 0.001
R(F²) = 0.05598	Background function: GSAS Background function number 1 with 2 terms. Shifted Chebyshev function of 1st kind.
1399 data points	Preferred orientation correction: General Simple Harmonic (GSH) description using Orientation Distribution Function (ODF) Von Dreele, R. B. (1997). J. Appl. Cryst. 30, 517–525.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq	Occ. (<1)
Hf	0.2751 (3)	0.0396 (3)	0.2073 (3)	0.0049 (7)	0.9962 (16)
O1	0.066 (4)	0.326 (4)	0.339 (4)	0.016 (6)
O2	0.456 (4)	0.757 (3)	0.487 (5)	0.007 (5)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

Geometric parameters (Å, º) top

Hf—Hfⁱ	3.441 (3)	Hf—O2ⁱ	2.16 (3)
Hf—Hfⁱⁱ	3.441 (3)	Hf—O2	2.21 (3)
Hf—Hf	3.305 (4)	Hf—O2	2.21 (3)
Hf—Hf	3.447 (4)	O1—Hf	2.01 (3)
Hf—Hf	3.424 (2)	O1—Hf^v	2.05 (3)
Hf—Hf	3.424 (2)	O1—Hf	2.18 (2)
Hf—O1	2.01 (3)	O2—Hf^vi	2.18 (3)
Hf—O1ⁱⁱⁱ	2.05 (3)	O2—Hfⁱⁱ	2.16 (3)
Hf—O1	2.18 (2)	O2—Hf	2.21 (3)
Hf—O2^iv	2.18 (3)	O2—Hf	2.21 (3)

O1—Hf—O1ⁱⁱⁱ	87.0 (4)	O2^iv—Hf—O2ⁱ	116.0 (6)
O1—Hf—O1	82.1 (6)	O2^iv—Hf—O2	71.1 (7)
O1—Hf—O2^iv	116.7 (9)	O2^iv—Hf—O2	74.1 (4)
O1—Hf—O2ⁱ	101.3 (8)	O2ⁱ—Hf—O2	74.5 (5)
O1—Hf—O2	72.6 (8)	O2ⁱ—Hf—O2	76.0 (7)
O1—Hf—O2	168.4 (8)	O2—Hf—O2	116.8 (6)
O1ⁱⁱⁱ—Hf—O1	77.2 (9)	Hf—O1—Hf^v	147.7 (10)
O1ⁱⁱⁱ—Hf—O2^iv	88.9 (8)	Hf—O1—Hf	109.3 (9)
O1ⁱⁱⁱ—Hf—O2ⁱ	145.0 (9)	Hf^v—O1—Hf	102.8 (9)
O1ⁱⁱⁱ—Hf—O2	139.6 (9)	Hf^vi—O2—Hfⁱⁱ	105.0 (12)
O1ⁱⁱⁱ—Hf—O2	89.1 (8)	Hf^vi—O2—Hf	108.9 (7)
O1—Hf—O2^iv	156.2 (8)	Hf^vi—O2—Hf	130.8 (10)
O1—Hf—O2ⁱ	70.6 (8)	Hfⁱⁱ—O2—Hf	103.1 (9)
O1—Hf—O2	131.3 (8)	Hfⁱⁱ—O2—Hf	104.0 (7)
O1—Hf—O2	86.4 (9)	Hf—O2—Hf	102.0 (11)

Symmetry codes: (i) −x+1, y−1/2, −z+1/2; (ii) −x+1, y+1/2, −z+1/2; (iii) −x, y−1/2, −z+1/2; (iv) x, y−1, z; (v) −x, y+1/2, −z+1/2; (vi) x, y+1, z.

Hafnium oxide (I_Neutron) top

Crystal data top

HfO₂	Z = 4
M_r = 210.49	D_x = 10.129 Mg m⁻³
Monoclinic, P2₁/c	Neutron radiation, λ = 1.48 Å
a = 5.1145 (2) Å	T = 295 K
b = 5.1682 (2) Å	Particle morphology: sphere
c = 5.2900 (2) Å	white
β = 99.211 (3)°	cylinder, 50 × 5 mm
V = 138.03 (1) Å³	Specimen preparation: Prepared at 295 K and 101.3 kPa

Data collection top

Powder diffractometer-3 (PD-3)	Data collection mode: transmission
Radiation source: nuclear reactor, Dhruva reactor, BARC	Scan method: Stationary detector
Si-crystal monochromator	Absorption correction: for a cylinder mounted on the φ axis GSAS Absorption/surface roughness correction: function number 0 (Lobanov & alte da Veiga, 1998).6th European Powder Diffraction Conference, 22-25 August 1998, Budapest, Hungary, Abstract P12-16.) Debye-Scherrer absorption correction Term (= MU.r/wave) = 0.78900 Correction is not refined.
Specimen mounting: vanadium can	T_min = 0.154, T_max = 0.220

Refinement top

Least-squares matrix: full	Profile function: CW Profile function number 2 Profile coefficients for Simpson's rule integration of pseudovoigt function Howard, C.J. (1982). J. Appl. Cryst., 15, 615–620. Thompson, P., Cox, D. E. & Hastings, J. B. (1987). J. Appl. Cryst., 20, 79–83. Peak tails are ignored where the intensity is below 0.0010 times the peak.
R_p = 0.024	48 parameters
R_wp = 0.031	0 restraints
R_exp = 0.017	(Δ/σ)_max = 0.05
R(F²) = 0.05598	Background function: GSAS Background function number 1 with 14 terms. Shifted Chebyshev function of 1st kind.
2421 data points	Preferred orientation correction: General Simple Harmonic (GSH) description using Orientation Distribution Function (ODF). Von Dreele, R. B. (1997). J. Appl. Cryst., 30, 517–525.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Hf	0.2747 (4)	0.0402 (4)	0.2078 (4)	0.006 (2)
O1	0.0668 (5)	0.3267 (4)	0.3420 (4)	0.0090 (5)
O2	0.4519 (5)	0.7551 (5)	0.4844 (5)	0.0073 (4)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

Geometric parameters (Å, º) top

Hf—Hfⁱ	3.444 (3)	Hf—O2ⁱ	2.163 (4)
Hf—Hfⁱⁱ	3.444 (3)	Hf—O2	2.235 (4)
Hf—Hf	3.305 (4)	Hf—O2	2.211 (4)
Hf—Hf	3.455 (5)	O1—Hf	2.017 (4)
Hf—Hf	3.420 (3)	O1—Hf^v	2.047 (4)
Hf—Hf	3.420 (3)	O1—Hf	2.166 (3)
Hf—O1	2.017 (4)	O2—Hf^vi	2.170 (4)
Hf—O1ⁱⁱⁱ	2.047 (4)	O2—Hfⁱⁱ	2.163 (4)
Hf—O1	2.166 (3)	O2—Hf	2.235 (4)
Hf—O2^iv	2.170 (4)	O2—Hf	2.211 (4)

O1—Hf—O1ⁱⁱⁱ	87.36 (9)	O2^iv—Hf—O2ⁱ	115.72 (11)
O1—Hf—O1	82.54 (10)	O2^iv—Hf—O2	71.65 (13)
O1—Hf—O2^iv	116.61 (14)	O2^iv—Hf—O2	74.28 (8)
O1—Hf—O2ⁱ	101.85 (13)	O2ⁱ—Hf—O2	73.94 (9)
O1—Hf—O2	72.53 (12)	O2ⁱ—Hf—O2	75.64 (13)
O1—Hf—O2	168.22 (14)	O2—Hf—O2	116.99 (11)
O1ⁱⁱⁱ—Hf—O1	76.67 (12)	Hf—O1—Hf^v	146.87 (12)
O1ⁱⁱⁱ—Hf—O2^iv	88.65 (12)	Hf—O1—Hf	109.67 (14)
O1ⁱⁱⁱ—Hf—O2ⁱ	144.97 (14)	Hf^v—O1—Hf	103.33 (12)
O1ⁱⁱⁱ—Hf—O2	140.37 (15)	Hf^vi—O2—Hfⁱⁱ	105.27 (13)
O1ⁱⁱⁱ—Hf—O2	88.48 (12)	Hf^vi—O2—Hf	108.35 (13)
O1—Hf—O2^iv	155.58 (13)	Hf^vi—O2—Hf	131.78 (16)
O1—Hf—O2ⁱ	71.26 (12)	Hfⁱⁱ—O2—Hf	102.08 (14)
O1—Hf—O2	131.37 (13)	Hfⁱⁱ—O2—Hf	104.36 (13)
O1—Hf—O2	85.77 (12)	Hf—O2—Hf	101.50 (12)

Symmetry codes: (i) −x+1, y−1/2, −z+1/2; (ii) −x+1, y+1/2, −z+1/2; (iii) −x, y−1/2, −z+1/2; (iv) x, y−1, z; (v) −x, y+1/2, −z+1/2; (vi) x, y+1, z.

Hafnium oxide (combined) top

Crystal data top

HfO₂	V = 138.03 (1) Å³
M_r = 210.49	Z = 4
Monoclinic, P2₁/c	D_x = 10.129 Mg m⁻³
a = 5.1145 (2) Å	Neutron/ synchrotron X-ray radiation
b = 5.1682 (2) Å	T = 295 K
c = 5.2900 (2) Å	Particle morphology: sphere
β = 99.211 (3)°	white

Data collection top

Powder
diffractometer-3 (PD-3)/ Measurement at synchrotron beamline

Radiation source: nuclear reactor/ synchrotron, Dhruva reactor, BARC/ Xpress beamline, Elettra

Refinement top

Least-squares matrix: full	0 restraints
R_p = 0.026	(Δ/σ)_max = 0.05
R_wp = 0.035	Background function: GSAS Background function number 1. Shifted Chebyshev function of 1st kind.
R_exp = 0.022	Preferred orientation correction: General Simple Harmonic (GSH) description using Orientation Distribution Function (ODF). Von Dreele, R. B. (1997). J. Appl. Cryst. 30, 517-525.
Profile function: CW Profile function number 2 Profile coefficients for Simpson's rule integration of pseudovoigt function Howard, C. J. (1982). J. Appl. Cryst., 15, 615–620. Thompson, P., Cox, D. E. & Hastings, J. B. (1987). J. Appl. Cryst., 20, 79–83. Peak tails are ignored where the intensity is below 0.0010 times the peak.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq	Occ. (<1)
Hf	0.2749 (3)	0.0402 (2)	0.2084 (3)	0.0066 (2)	0.9979 (17)
O1	0.0670 (5)	0.3273 (4)	0.3428 (4)	0.0099 (4)
O2	0.4512 (4)	0.7558 (4)	0.4840 (4)	0.0074 (3)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

Geometric parameters (Å, º) top

Hf—Hfⁱ	3.442 (2)	Hf—O2ⁱ	2.167 (3)
Hf—Hfⁱⁱ	3.442 (2)	Hf—O2	2.234 (3)
Hf—Hf	3.310 (3)	Hf—O2	2.215 (3)
Hf—Hf	3.458 (4)	O1—Hf	2.020 (3)
Hf—Hf	3.420 (2)	O1—Hf^v	2.047 (3)
Hf—Hf	3.420 (2)	O1—Hf	2.164 (3)
Hf—O1	2.020 (3)	O2—Hf^vi	2.163 (3)
Hf—O1ⁱⁱⁱ	2.047 (3)	O2—Hfⁱⁱ	2.167 (3)
Hf—O1	2.164 (3)	O2—Hf	2.234 (3)
Hf—O2^iv	2.163 (3)	O2—Hf	2.215 (3)

O1—Hf—O1ⁱⁱⁱ	87.41 (7)	O2^iv—Hf—O2ⁱ	115.78 (8)
O1—Hf—O1	82.59 (9)	O2^iv—Hf—O2	71.61 (10)
O1—Hf—O2^iv	116.62 (11)	O2^iv—Hf—O2	74.36 (6)
O1—Hf—O2ⁱ	101.76 (11)	O2ⁱ—Hf—O2	73.88 (7)
O1—Hf—O2	72.60 (10)	O2ⁱ—Hf—O2	75.77 (10)
O1—Hf—O2	168.10 (11)	O2—Hf—O2	117.05 (8)
O1ⁱⁱⁱ—Hf—O1	76.39 (11)	Hf—O1—Hf^v	146.61 (12)
O1ⁱⁱⁱ—Hf—O2^iv	88.80 (10)	Hf—O1—Hf	109.64 (12)
O1ⁱⁱⁱ—Hf—O2ⁱ	144.79 (11)	Hf^v—O1—Hf	103.61 (11)
O1ⁱⁱⁱ—Hf—O2	140.60 (12)	Hf^vi—O2—Hfⁱⁱ	105.30 (10)
O1ⁱⁱⁱ—Hf—O2	88.30 (10)	Hf^vi—O2—Hf	108.39 (10)
O1—Hf—O2^iv	155.48 (11)	Hf^vi—O2—Hf	132.00 (12)
O1—Hf—O2ⁱ	71.32 (10)	Hfⁱⁱ—O2—Hf	101.98 (11)
O1—Hf—O2	131.51 (10)	Hfⁱⁱ—O2—Hf	104.23 (10)
O1—Hf—O2	85.62 (10)	Hf—O2—Hf	101.37 (9)

Symmetry codes: (i) −x+1, y−1/2, −z+1/2; (ii) −x+1, y+1/2, −z+1/2; (iii) −x, y−1/2, −z+1/2; (iv) x, y−1, z; (v) −x, y+1/2, −z+1/2; (vi) x, y+1, z.

Comparative study of structural parameters of monoclinic HfO₂ (P2₁/c) from previous X-ray diffraction studies top

Cell parameters	a (Å)	5.1187 (4)	5.1170 (1)	5.1156 (5)
	b (Å)	5.1693 (1)	5.1754 (2)	5.1722 (5)
	c (Å)	5.2970 (4)	5.2915 (2)	5.2948 (5)
	β (°)	99.180 (9)	99.216 (2)	99.18 (8)
Volume (Å³)	V		138.36 (2)	136.441 (2)	138.30 (8)
Atomic positions/	Hf	x	0.2764 (2)	0.2755 (2)	0.2759 (5)
displacement parameters		y	0.0402 (1)	0.0397 (1)	0.0412 (5)
		z	0.2074 (2)	0.2080 (2)	0.2078 (5)
		B_iso (Å²)	0.34 (1)	0.05 (2)	0.45 (5)
	O1	x	0.0709 (20)	0.0742 (21)	0.073 (8)
		y	0.3319 (18)	0.3316 (18)	0.346 (8)
		z	0.3438 (18)	0.3467 (17)	0.332 (8)
		B_iso (Å²)	1.52 (1)	0.4 (3)	0.45 (5)
	O2	x	0.4464 (17)	0.4487 (21)	0.446 (8)
		y	0.7554 (16)	0.7581 (16)	0.748 (8)
		z	0.4796 (18)	0.4801 (23)	0.488 (8)
		B_iso (Å²)	0.63 (3)	0.8 (4)	0.45 (5)
Interatomic	Hf—O1		2.036 (11)	2.031 (11)	2.05 (18)
distance (Å)			2.059 (10)	2.069 (11)	2.03 (5)
			2.146 (10)	2.128 (9)	2.2 (4)
	Hf—O2		2.163 (10)	2.170 (12)	2.17 (5)
			2.146 (9)	2.139 (10)	2.20 (5)
			2.210 (10)	2.227 (11)	2.16 (5)
			2.264 (9)	2.250 (11)	2.25 (4)
Interatomic	Hf—O1—Hf		109.9 (5)	110.9 (5)	108 (8)
angle (°)	Hf—O2—Hf		100.3 (4)	100.5 (5)	102.30 (17)
			101.2 (4)	101.6 (5)	101.10 (17)
			103.8 (4)	103.4 (4)	105.50 (18)
			105.7 (5)	106.0 (6)	103.70 (19)
R (%)					10.1
R_wp (%)			10.91	9.61
Exp R_wp (%)			5.19	7.05

References: (a) Adams et al., (1991); (b) Hann et al. (1985); (c) Ruh & Corfield (1970).

Crystal structure parameters, a.d.p.'s and goodness-of-fit parameters of monoclinic bulk HfO₂ (P2₁/c) from the individual refinement of synchrotron XRD (at λ = 0.5007 Å) pattern, ND pattern (at λ = 1.48 Å) and the combined refinement with both sets of data top

Lattice	a		5.11447 (21)
parameters (Å)	b		5.16819 (24)
	c		5.29000 (24)
	β		99.2109 (33)
Volume (Å³)	V		138.025 (8)'
Atomic	Hf	x	0.27514 (33)	0.27470 (40)	0.27490 (28)
Positions/		y	0.03964 (28)	0.04020 (35)	0.04019 (24)
Isotropic Atomic		z	0.20728 (29)	0.20780 (40)	0.20838 (26)
Displacement		U_iso (Å²)	0.0049 (7)	0.0063 (21)	0.0066 (2)
Parameters		B_iso (Å²)	0.39 (5)	0.50 (17)	0.52 (1)
	O1	x	0.0660 (40)	0.0668 (5)	0.0670 (5)
		y	0.3260 (40)	0.3267 (4)	0.3273 (4)
		z	0.3390 (35)	0.3420 (4)	0.3428 (4)
		U_iso (Å²)	0.0160 (60)	0.0090 (5)	0.0099 (4)
		B_iso (Å²)	1.26 (47)	0.71 (4)	0.78 (3)
	O2	x	0.4560 (40)	0.4519 (5)	0.4512 (4)
		y	0.7568 (29)	0.7551 (5)	0.7558 (4)
		z	0.4870 (50)	0.4844 (5)	0.4840 (4)
		U_iso (Å²)	0.0070 (50)	0.0073 (4)	0.0074 (3)
		B_iso (Å²)	0.55 (39)	0.58 (3)	0.59 (3)
Interatomic	Hf—O1		2.01 (3)	2.017 (4)	2.020 (3)
distance (Å)			2.05 (3)	2.047 (4)	2.047 (3)
			2.18 (2)	2.166 (3)	2.164 (3)
	Hf—O2'		2.16 (3)	2.163 (4)	2.167 (3)
			2.21 (3)	2.047 (4)	2.163 (3)
			2.21 (3)	2.235 (4)	2.234 (3)
Interatomic	Hf—O1—Hf		109.3 (9)	109.67 (14)	109.64 (12)
angle (°)	Hf—O2—Hf		102.0 (11)	101.50 (12)	101.37 (9)
			103.1 (9)	102.08 (14)	101.98 (11)
			104.0 (7)	104.36 (13)	104.23 (10)
			105.0 (12)	105.27 (13)	105.30 (10)
Quality-of-fit parameters	R_wp (%)		11.10	3.09	11.50 (XRD pattern), 3.1 (ND pattern)
Reduced	χ²'		1.486	3.20	2.71

Optimized crystal structure parameters and electronic band gap of monoclinic bulk HfO₂ (P21/c) from first-principles density functional theory (DFT) top

Method	Lattice parameters (Å) and angle β (°)	Unit-cell volume (f.u. = 4) (Å³)	Hf—O bond length (Å)	Hf—O1—Hf/Hf—O2—Hf angles (°)	Band gap (eV) 5.68[D]^a, 5.77[D]^b, 5.1[I]^c
PBE-GGA (optimized)	a = 5.164, b = 5.192, c = 5.351, β = 99.008	141.67(+2.6%)	Hf—O1 (×3): 2.052, 2.059, 2.160; Hf—O2 (×4): 2.181, 2.148, 2.250, 2.283	110.905/100.052, 102.003,104.488, 104.488	4.69[D], 4.33[I]
PBE-GGA (optimized, previous data)^d	a = 5.215, b = 5.293, c = 5.350, β = 99.73	145.56(+5.5%)	NA	NA	3.6[I]
PBE-GGA (optimized, previous data)^e	a = 5.143, b = 5.190, c = 5.330, β = 99.6	NA	NA	NA	4.08[I]
B3LYP (optimized)	a = 5.177, b = 5.177, c = 5.362, β = 98.891	141.97(+2.9%)	Hf—O1 (×3): 2.051, 2.059, 2.157; Hf—O2 (×4): 2.181, 2.146, 2.251, 2.300	111.658/99.800, 102.007, 105.040, 105.040	6.59[D], 6.23[I]

Notes: f.u. is the formula unit. (a) D is the direct band gap from UV–Vis spectroscopy measurements on films and NA is not available (Balog et al., 1977); (b) D is the direct band gap from spectroscopy ellipsometry measurements on films (He et al., 2007); (c) I is the indirect band gap from optical absorption spectral measurements on nanocrystals (Padma Kumar et al., 2015); (d) calculations based on plane-wave norm-conserving pseudopotentials (Kang et al., 2003); (e) calculations (Ondračka et al., 2016)based on plane-waves and full potentials using the Wien2K code (Blaha et al., 2019).

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