Phonon frequencies in strained crystals

Figure 4.4: Molecular dynamics simulation of the axial (closed circles) and the circumferential (open circles) strain under hydrostatic pressure. The data points are taken from Fig. 4 of Ref. 29. The cir-cumferential deformation shown here corresponds to the average of the two radii under pressure re-ported by Venkateswaran et al.²⁹

0 1 2 3 4 5

0 5 10

15 ε_θθ = -3.41 TPa^-1p ε_zz = -0.91 TPa^-1p

molecular dynamics calculation

-Strain (10-3 )

Pressure (GPa)

value, whereas the circumferential strain was found to be much larger than in the continuum approximation.

Despite some differences in the absolute values of the predicted strains all four calculation agree in the following fundamental points: Under hydrostatic pressure the circumferential strain is by a factor of 2-4 larger than the axial strain. The linear compressibility along the nanotubes axis is similar to graphite (−0.8 TPa⁻¹p).

following how to derive the vibrational frequencies in nanotubes under pressure by making reference to the rolled up graphene sheet.

The basic idea of the approach is depicted in Fig. 4.5. The figure shows schematically a (6,6) and an (8,4) tube under exaggerated hydrostatic pressure (corresponding to≈100 GPa). Be-cause of the larger strain in the radial than in the axial direction, hydrostatic pressure changes not only the area of the graphene hexagons but also distorts their shape. The sixfold hexag-onal symmetry is broken under pressure, which splits the doubly degenerate E2g graphene optical modes into a higher and lower frequency component vibrating parallel and perpen-dicular to the higher strain direction, respectively. For the nanotubes this corresponds to a stronger pressure dependence for a phonon eigenvector where the atomic displacement is along the circumferential direction than for an axial vibration.

For a quantitative analysis I unwrap the tube to a narrow graphene rectangle. The strain in the graphene sheet due to ε_θθ ^andεzz reads (after transformation to the principle axis of graphene)⁹⁹

ε⁼εθθcos²Θ+εzzsin²Θ ¹₂sin(2Θ)(εzz−εθθ)

1

2sin(2Θ)(εzz−ε_θθ) ε_θθ^sin2Θ+εzzcos²Θ

(4.11) separated into the hydrostatic and non-hydrostatic components

= 1

2(ε_θθ+εzz) 1 0

0 1

+1

2(ε_θθ−εzz)

cos 2Θ sin 2Θ sin 2Θ −cos 2Θ

; (4.12)

where Θ is the chiral angle. Obviously, the deformation of the graphene elementary cell given in Eq. (4.11) is not purely hydrostatic for any chirality.

To find the phonon frequencies in the strained graphene cell I expand the dynamical equation to terms linear in strain^{114, 117}

−m ¨v_i=mω0²v_i+

∑

klm

K_ikml⁽¹⁾ εlmv_k; (4.13)

where vvv is the atomic displacement, m the reduced mass of the atoms, andω0the strain free frequency. The second summand describes the change in phonon frequency due to the strain;

Figure 4.5: Schematic picture of the distortion of a (6,6) and an (8,4) nanotube under hydrostatic pres-sure, i.e.,εθθ=2εzz. The strain is fully symmetric in the point groups of the tubes, but not for the graphene hexagon. The hexagon area as well as its shape are altered by applying pressure to a nan-otubes.

(6,6) (8,4)

K_ikml=∂^Kik/∂εklis the change in the spring constants of the strained crystal. The symmetric tensor K⁽¹⁾ has only three nonzero components because of the hexagonal symmetry of the graphene sheet, namely,¹⁰³

K₁₁₁₁=K₂₂₂₂ = m·K˜₁₁

K₁₁₂₂ = m·K˜₁₂ (4.14)

K₁₂₁₂ = m·1

2(K˜₁₁−K˜₁₂).

From the dynamical equation (4.13) I obtain an secular equation with the help of the K⁽¹⁾ tensor components in Eq. (4.14) and the strain in the graphene sheet (4.11)^{114, 117}

1 2

(K˜₁₁+K˜₁₂)(εθθ+εzz)+

(K˜₁₁−K˜₁₂)(εθθ−εzz)sin 2Θ +(K˜₁₁−K˜₁₂)(εθθ−εzz)cos 2Θ−2λ

(K˜₁₁−K˜₁₂)(εθθ−εzz)sin 2Θ (K˜₁₁+K˜₁₂)(εθθ+εzz)−

−(K˜₁₁−K˜₁₂)(εθθ−εzz)cos 2Θ−2λ

≡ 0,

(4.15) whereλ =ω²₋ω₀²is the difference between the squared strain dependent frequencyω ^and the frequency in the absence of strainω0. Diagonalizing Eq. (4.15) thus yields the relative shift of the phonon energy in the strained graphene sheet.⁹⁹

ω₋ω0

ω0

= ∆ω ω0 ≈ λ

2ω₀^{2 =}

(K˜₁₁+K˜₁₂)

4ω₀^{2 (}ε_θθ+εzz)±1 2

(K˜₁₁−K˜₁₂)

2ω₀^{2 (}ε_θθ−εzz). (4.16) In Eq. (4.16) two phonon deformation potentials relate the frequency shift with strain. The first deformation potential (K˜11+K˜12)/4ω₀² =−γ is the Gr¨uneisen parameter, which de-scribes the frequency shift for an hydrostatic deformation of the graphene hexagon. The splitting of the modes under shear strain comes from the second term.

Interestingly, the frequency shift is independent of the chirality of nanotube, i.e., the way I cut and strained the graphene rectangle. A mode vibrating parallel toε_θθ, the high-strain di-rection, always shifts according to the plus solution of Eq. (4.16), whereas vibrations parallel toεzz have a frequency shift below the hydrostatic contribution. Between these two limiting cases the phonon modes show a dispersion similar to, e.g., the dispersion in wurzite crystals.

The dependence of the phonon frequency on the displacement direction of the eigenvector in strained materials is discussed in two papers by Anastassakis.^{118, 119}

4.3.1. Pressure dependence of the phonons frequencies in nanotubes

The discussion of the last section was restricted to the unwrapped nanotube and the resulting nonhydrostatic deformation of graphene. I now roll up the sheet again bearing in mind the

phonon energies in Eq. (4.16). For armchair tubes – and achiral tubes in general – I found in Chapter 2. the complete set of phonon eigenvectors by group projection operators and some general assumptions on the strength of the force constants. The Raman active high-energy A_1g (₀A⁺₀) and E_2g(₀E₂⁺) phonons are vibrating along the circumference in armchair tubes, while the E1g (0E₁⁻) mode has a purely axial eigenvector. The two circumferential vibrations are according to Eq. (4.16) expected to have a pressure derivative higher than the axial E_1g mode by(K˜₁₁−K˜₁₂)(εzz−εθθ)/2ω₀². In contrast, I presented in Section 4.1.

Raman experiments where the pressure slope was the same for all high-energy peaks. Before discussing this apparent contradiction I briefly turn to metallic nanotubes, which indeed show the expected splitting in experiment.

In Fig. 4.6 I present the phonon frequencies of metallic nanotubes as a function of applied hy-drostatic pressure. The insets show the Raman spectra at a low (0.34 GPa) and high-pressure (9.9 GPa) point. The intensity decrease, as in semiconducting nanotubes, is observed in metallic tubes as well. The analysis of the data is difficult due to the close proximity of the modes. Additionally, under pressure, the broadening of the peaks leads to a merging of the lines (see insets in Fig. 4.6). Nevertheless, when fitting the high-energy peak with three similarly sized Lorentzians or with a focus on the low and high-energy flanks for pressures up to 10 GPa, I find a splitting of 0.56 TPa⁻¹between the peaks at 1544 and 1565 cm⁻¹and a hydrostatic shift of 3.8 TPa⁻¹. The third peak, at 1592 cm⁻¹, which has been assigned to semiconducting tubes, has a pressure derivative of 3.8 TPa⁻¹.

For the comparison between the experiments and the theory developed in the last two sec-tions I need to know the two phonon deformation potentials in Eq. (4.16). Table 4.3 lists a compilation of measured and calculated results in carbon based material. The shear de-Figure 4.6: High-pressure Raman scattering on

metallic nanotubes excited with an excitation en-ergy 1.91 eV. The phonon energies were normal-ized to the value at ambient pressure. Up triangles refer to the mode at 1565 cm⁻¹ at zero pressure, down triangles to the one at 1544 cm⁻¹, and the open squares to the 1592 cm⁻¹ mode. The latter mode is usually assigned to semiconducting nan-otubes at this excitation energy.⁴⁴ The insets show Raman spectra (Raman shift in cm⁻¹) at p=0.34 and 9.9 GPa and the fit to the data.

1500 1600 1700

9.9 GPa

0 2 4 6 8 10 12

0 10 20 30 40

∆ω/ω 0 (10-3 )

Pressure (GPa)

1500 1600

S. Reich et al. Figure 2

0.34 GPa

γ ^{(K˜11−K˜12)} 2ω₀² nanotubes^{33, 76} 1.24 0.41 graphite 1.59¹¹³ 0.66¹²⁰ diamond (cubic)¹²¹ 1.50 0.51

Table 4.3: Phonon deformation potentials for nan-otubes (tight-binding calculation), graphite (exper-iment and ab-initio calculation), and diamond (ex-periment). The shear deformation potential in di-amond is for an applied uniaxial stress along the (001) direction.

formation potentials of graphene or graphite are not accessible experimentally, because of graphite’s brittleness when an uniaxial stress is applied along the basal plane. Using the circumferential and axial strain components obtained from elasticity theory and the elastic constant calculations by Lu¹⁰⁹together with the phonon deformation potentials for graphene and nanotubes I find an hydrostatic component γ·(εzz+εθθ) =−(3.8±1)TPa⁻¹p and a shear strain splitting(K˜₁₁−K˜₁₂)(εzz−εθθ)/2ω₀²=−(0.6±0.2)TPa⁻¹p. Both values are in very good agreement with the experimental values for metallic nanotubes. The large uncertainty is due to the uncertainties in the strain determination as well as in the phonon deformation potentials. Precise measurements of the elastic properties of carbon nanotubes are desirable to further clarify the high-pressure measurements.

The pressure dependence of the Raman peaks in metallic nanotubes is in general agreement with the idea of predominantly axial and circumferential high-energy Raman active vibra-tions. On the other hand, in semiconducting tubes only the hydrostatic part of the pressure shift was observed experimentally. The possible explanation for the absence of the shear strain splitting lies in the phonon eigenvectors of chiral nanotubes. In contrast to the higher-symmetry achiral tubes the phonon eigenvectors of chiral tubes are not fixed by higher-symmetry because of the missing mirror planes, see Section 2.4. A distribution of displacement direc-tions with respect to the circumference or the tube axis washes out the splitting introduced by the shear deformation in Eq. (4.16).^{76, 118} Only the hydrostatic component is then ob-served experimentally. To find out how the eigenvectors of chiral tubes actually look like I calculated the phonons of two selected chiral and two achiral nanotubes by first principles calculation.

Im Dokument Carbon Nanotubes: Vibrational and electronic properties (Seite 79-83)