Jose A.
Alarco
*ab,
Alison
Chou
ab,
Peter C.
Talbot
ab and
Ian D. R.
Mackinnon
a
aInstitute for Future Environments, Queensland University of Technology, 2 George St., Brisbane, QLD, 4001 Australia. E-mail: jose.alarco@qut.edu.au
bScience and Engineering Faculty, Queensland University of Technology, 2 George St., Brisbane, QLD, 4001 Australia
First published on 25th September 2014
Micrometre-sized MgB2 crystals of varying quality, synthesized at low temperature and autogenous pressure, are compared using a combination of Raman and infra-red (IR) spectroscopy. These data, which include new peak positions in both spectroscopies for high quality MgB2, are interpreted using DFT calculations on phonon behaviour for symmetry-related structures. Raman and IR activity additional to that predicted by point group analyses of the P6/mmm symmetry are detected. These additional peaks, as well as the overall shapes of calculated phonon dispersion (PD) models are explained by assuming a double super-lattice, consistent with a lower symmetry structure for MgB2. A 2× super-lattice in the c-direction allows a simple correlation of the pair breaking energy and the superconducting gap by activation of corresponding acoustic frequencies. A consistent physical interpretation of these spectra is obtained when the position of a phonon anomaly defines a super-lattice modulation in the a–b plane.
We have recently synthesized MgB2 under conditions that result in a reproducible, dense and coarse-grained microstructure.28 This coarse-grained MgB2 has excellent electrical, magnetic and transport properties. By varying conditions of this synthesis process, we have also produced inferior quality MgB2 which shows a lower superconducting transition temperature (Tc = 38.0 K) and lower density, finer-grained material.28 This latter type of material is similar in form and property to that for which several spectroscopic observations on MgB2 are reported.5,13,15 This work examines differences in optical properties of two different forms of MgB2, in order to evaluate potential causes for disparity of results in the current literature.1,2,18–20,25–27
Density functional theory (DFT) is used to estimate the phonon behaviour of MgB2 based on well-determined crystallographic data using neutron and X-ray diffraction.29–31 Point group analysis of the P6/mmm symmetry for MgB2 predicts one Raman and two IR active peaks (see Table 4). However, other studies1,2,8,15,32,33 as well as this one, suggest that additional Raman or IR active peaks are observed. Ab initio DFT has been used to calculate the phonon density of states (PDOS) and the phonon dispersion (PD) of MgB2.21,32,34–36 However, this literature also appears to show somewhat divergent results between computational models and experimental data.1,32,34–36 In this work, experimental data are considered in light of modelling to estimate phonon frequencies, density of states and dispersion relations in MgB2 with a particular focus on high quality MgB2 crystals.
For these Raman experiments, the samples, while still in the glove box, are placed inside a gas-tight Teflon sample chamber fitted with an optical (glass) window for Raman measurements. For IR measurements, the samples are handled in a similar manner to reduce exposure to air at all times up to the initiation of data collection. We have evaluated samples degraded in air over long times (e.g. days) to confirm Raman and IR characteristics of these materials. Data reported in this work are for samples with minimal degradation. Two types of MgB2 samples are produced by the reactions described by Mackinnon et al.28 and a summary of essential characteristics is given in Table 1.
Characteristics | Sample A | Sample B |
---|---|---|
Maximum temp. of reaction (°C) | 500 | 500 |
Heating profile hold temp. (°C) | At 50 and 250 | No hold |
Superconducting temp., Tc (°C) | 38.5 | 38.0 |
Particle size (μm) | 40–80 | 5–10 |
Particle morphology | Euhedral hexagon or bipyramidal | Thin platelet discs |
Dominant color (incident light) | Gold | Black/grey |
Lustre | Bright | Dull |
Raman spectra are collected with an inVia Renishaw Raman microscope using two different excitation wavelengths: (i) a near-IR diode laser operating at 785 nm or (ii) a He–Ne laser operating at 532 nm. The laser power at the sample is 20 mW for the 785 nm laser and 10 mW for the 532 nm laser, focussed into a spot size of ∼1 μm through a ×50 objective lens. Each Raman spectrum consists of 4 accumulations with a 20 s exposure per scan. Spectrum noise and background are corrected using the smoothing and baseline correction functions in the GRAMS software.37
Attenuated total reflectance (ATR) FTIR measurements are recorded with a Nicolet iS50 ATR-FTIR spectrophotometer internally purged with dry air. All measurements are performed at room temperature with the sample (exposed to air) in direct contact with a diamond ATR crystal. Data are collected over an accumulation of 8000 scans at a speed of 0.02 min per scan, energy step 0.428 cm−1 and resolution of 4 cm−1 over the range of 350–1800 cm−1.
A Leica multi-focus, stereo optical microscope and a Zeiss Sigma variable pressure field emission SEM with Oxford instruments silicon drift detector (SDD) are used for microscopy observations and energy dispersive spectroscopy (EDS) elemental analysis. Samples are prepared for SEM/EDS by placing a thin layer of powder onto aluminium stubs with double-sided carbon tape. In general, samples are not coated with a conductive coating to avoid analytical interference(s).
Most calculations are completed using Materials Studio 7.0 via the Microsoft Windows mode of a 12 core Mac Pro Xeon 64 bit workstation. This workstation operates with an Intel chip and mimics up to 24 cores. When memory requirements exceed the capacity of this computer, calculations are undertaken via the high performance computing (HPC) facility at QUT using multiples of 16 cores. The optimum choice for calculations on the HPC facility is 64 cores.
Specific functionals and the acronyms used to describe functionals in these CASTEP calculations (e.g. LDA–CA–PZ or GGA–PBE) for a different, albeit cubic, structure are described by Alarco et al.40 The DFPT method is implemented in CASTEP for norm conserving potentials (NCPs). On the other hand, FD methods, although in principle available for both NCPs and ultrasoft potentials (UPs) are more effective with UPs. Some FD calculations using NCPs are possible with coarse k-grids but use of a finer k-grid resulted in negative frequency values and difficulty with convergence.
For single cell calculations on MgB2, the lattice parameters a = b = 0.3085 nm and c = 0.3523 nm have been used as input.29–31 For super-lattice calculations, appropriate multiples of these parameters have been used. Fig. 1 shows a schematic of the reciprocal space lattice for P6/mmm symmetry of MgB2 where the green and grey spheres represent Mg and B atoms, respectively. Super-lattice constructions of the MgB2 cell used in this work are multiples along the c-axis direction that retain similar reciprocal space directions.
Fig. 1 Reciprocal space projection for the P6/mmm group that shows major symmetry directions. The real space primitive cell is also depicted. |
SEM images of Samples A and B are shown in Fig. 2. The difference in morphology between each sample is demonstrable albeit each sample shows a similar X-ray diffraction pattern for MgB2.28
Fig. 2 SEM images of MgB2 made by the same process with different operating parameters as shown in Table 1. (a) Sample A showing large euhedral crystal shapes approx. 40 μm to 60 μm and (b) Sample B showing smaller sized platy discs approx. 5 μm to 10 μm. |
Fig. 3b shows a comparison of Sample A and B under the same illuminating conditions. These spectra show that larger size grains enhance the observation of well-defined Raman peaks as noted in earlier work on different compounds by a number of authors.41–43,47
Fig. 4 shows an example of Raman spectra from Sample A acquired at 532 nm with freshly prepared material and minimum exposure to air. Notice the peak at ∼230 cm−1, which is close to peaks observed and calculated in other Raman studies of MgB2.2,11,32,34 Additional Raman peaks (labelled with “*”) are also observed in Fig. 4b, presumably due to different orientations of these larger size grains. Minor amplitude variations due to instrument noise are also observed in Fig. 4b (arrowed).
Fig. 5 shows Raman spectra for Sample A collected on different days, at two laser excitations. The spectra show a close match of peak positions at low wavenumbers. Up to nineteen Raman active peaks as compiled in Table 2 can be identified in the spectra collected on this high quality sample of MgB2.
Peak | Peak centre [cm−1] | |||
---|---|---|---|---|
From Fig. 3a | From Fig. 4 | From Fig. 5 | From Fig. 5 | |
532 nm | 785 nm | |||
a Laser power at 50% produces slightly higher peak shift positions compared with 100% laser power. b This position is uncertain due to over-lapping intensity at a shoulder peak. | ||||
1 | 230 | 246 | 247 | |
2 | 299 | 300b | 297 | |
3 | 353 | |||
4 | 382 | 393 | 397 | 398 |
5 | 406 | 406 | ||
6 | 464 | 465 | 454 | |
7 | 530–552 | |||
8 | 603 | 590 | 594 | |
9 | 657 | |||
10 | 767 | 767 | 749 | |
11 | 803 | 800 | ||
12 | 908 | 906 | ||
13 | 940–973 | 958 | ||
14 | 1047 | |||
15 | 1062–1073 | |||
16 | 1295 | |||
17 | 1385 | 1378 | ||
18 | 1435 | |||
19 | 2433 |
Table 2 gives a summary of the peak positions detected in the Raman spectra from Sample A as shown in Fig. 3–5. Inspection of data in Table 2 shows that some frequencies – within experimental error – are grouped together or are common to both illumination wavelengths. Minor differences in explicit values may be attributed to slightly different temperature effects. However, for experiments summarized in Table 2, low energy peaks consistently occur and occur repeatedly with comparable intensity and peak width. This occurrence implies that these low energy peaks are active to similar extent and are not governed by P6/mmm symmetry.
Table 3 lists the approximate peak positions of IR spectra for samples A and B. For sample A, peak parameters are determined by Gaussian peak-fitting routines with GRAMS software.37 For sample B, peak positions are listed since Gaussian fits do not produce reasonable spectral shapes. This spectrum is also difficult to fit using combinations of Lorentzian and Gaussian peak shapes.
Sample A | Sample B | ||||
---|---|---|---|---|---|
Peak No. | Centre [cm−1] | Height | Width [cm−1] | Area | Centre [cm−1] |
1 | 384 | 0.0104 | 36 | 0.39 | 385, 394 |
2 | 425 | 0.0101 | 45 | 0.48 | 412, 427 |
3 | 557 | 0.0078 | 96 | 0.80 | 547, 570 |
4 | 628 | ||||
5 | 820 | ||||
6 | 868 | 0.0033 | 110 | 0.39 | 878 |
7 | 930 | 0.0017 | 48 | 0.09 | 928 |
8 | 990 | 0.0043 | 80 | 0.37 | |
9 | 1053 | ||||
10 | 1190 | 0.0017 | 103 | 0.18 | 1183 |
11 | 1282 | 0.0033 | 156 | 0.55 | |
12 | 1369 | ||||
13 | 1428 | 0.0047 | 136 | 0.69 | 1427 |
14 | 1632 | 0.0016 | 92 | 0.16 | 1604 |
15 | 1727 | 0.0016 | 114 | 0.18 | |
16 | 1804 |
The region between 400 cm−1 and 650 cm−1 around the G (or Γ) point is a noteworthy feature (circled in Fig. 7) that varies considerably with change in k-grid interval. For Raman active frequencies, the region around the G direction shifts from a single degenerate “dip” at ∼450 cm−1 at the G centre point to a degenerate “Mexican hat” configuration (with two adjacent dips) located between the G–K and G–M directions with decreasing grid interval. In addition, the doublet minima, or approximately degenerate bands, shift(s) to higher frequency at ∼550 cm−1.
Table 4 shows the values for Raman active frequencies as the interval of the grid is decreased. A substantial change from 425 cm−1 to the 550–650 cm−1 range occurs with decreasing k-grid interval. These frequencies span the range of values reported in the literature for MgB2 under different experimental conditions as noted below.
Minimum k-vector [A−1] | Raman active mode [cm−1] |
---|---|
0.04 | 425.89 |
0.03 | 644.61 |
0.02 | 582.29 |
0.015 | 551.04 |
Similar calculations have been attempted using the FD method, but convergence occurs only with use of UPs. For example, with a coarse k-grid at 0.07 A−1, anomalous behaviour of the lower phonon bands occurs along the reciprocal space AH–HK and ML–LH directions. This anomalous behaviour becomes extreme (i.e. with negative values), when NCPs are used with either LDA–CA–PZ or GGA–PBE functionals. However, calculations using UPs eliminate these lower band anomalies in most cases. A strong peak at a very high frequency (∼890–910 cm−1) for the Raman active mode at the Γ (or G) point is maintained at fine grid scale.
Additional calculations with the FD method show that the lowest acoustic band tends to deviate from other acoustic bands when using the GGA–PBE functional and is less pronounced for the LDA–CA–PZ functional. This relative difference occurs for fine k-grids of 0.02 A−1 and is shown in Fig. 8 as PD plots. However, the peaked Raman active mode reverses shape to form a “valley” near the G point. This form of the Raman active mode is similar to that determined from DFPT methods with corresponding calculated frequency values closer to experimental values (compare Fig. 7 and 8). Optimum results for the FD method are obtained with this k-grid interval and a super-cell cut-off radius of 5 A.
Fig. 8 Phonon dispersion (PD) for space group P6/mmm calculated with the FD method using k = 0.02 A−1 and a super-cell cutoff radius 5 A for (a) LDA–CA–PZ functional and (b) GGA–PBE functional. |
Calculations using a larger cut-off radius and/or a finer k-grid to improve accuracy led to unviable, extremely long calculation times. These calculations are terminated once the pre-set time limit between three and five days is achieved without iteration progress.
Frequency (cm−1) | Group | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
P6 | P | P6/m | P622 | P6mm | Pm2 | P6/mmm | ||||||||
168b | 174b | 175b | 177b | 183b | 187b | 191b | ||||||||
IR | R | IR | R | IR | R | IR | R | IR | R | IR | R | IR | R | |
a These values display strong dependence on the dimension of the k-grid. Other frequencies display smaller variations (<6 cm−1) that are largely attributable to minor differences in lattice parameters calculated during geometry optimization. (dd) indicates a doubly degenerate mode. b Value is the space group number. | ||||||||||||||
342.8–344.3 (dd) | Y | Y | Y | Y | Y | N | Y | Y | Y | Y | Y | Y | Y | N |
413.3–416.9 | Y | Y | Y | N | Y | N | Y | N | Y | Y | Y | N | Y | N |
425.9–644.7a (dd) | N | Y | Y | Y | N | Y | N | Y | N | Y | Y | Y | N | Y |
711.6–716.9 | N | N | Y | N | N | N | N | N | N | N | Y | N | N | N |
In this work, examples of lower symmetry which cannot be represented in a single unit cell have also been explored. Thus, to achieve the distribution of atoms, we have used super-lattice structures that reproduce the spatial arrangement of Mg and B atoms in the basic MgB2 structure.29–31 For the double super-lattice construction in the c-direction to represent the primitive cell, a lower symmetry and separate insertion of two B atoms is required. This construct is not necessary for the higher P6/mmm symmetry, where inserting one B atom replicates all the other B atom positions. Thus, in this format where B atoms are linked by different symmetry elements compared to the basic P6/mmm symmetry of MgB2,29–31 movement of adjacent B atoms is less constrained in order to conserve symmetry. A similar situation can be envisaged for Mg atoms under an appropriate choice of reduced symmetry as shown for Pc1 and can be constructed for the other forms of lower symmetry described in this work.
Fig. 9 shows the PD as a function of decreasing interval of k-grid for the lower group symmetry Pc1. The broad features of the PD for P6/mmm symmetry as shown in Fig. 7 and 8 are largely maintained albeit a higher band population along similar reciprocal directions appears. The “Mexican hat” form of the near degenerate bands in the region 500–600 cm−1 still remains. At the smaller k-grid interval (Fig. 9b), an additional band at ∼650 cm−1 occurs but is not bifurcated nor a doublet. Fig. 9b also shows that the optical and acoustic bands are in closer proximity compared with Fig. 9a at higher k-grid interval. As expected, along the super-lattice ΓA line direction, substantial creation of new branches and/or breaking of degeneracy can be observed as a result of the folding of the Brillouin zone (in similar fashion to illustrations in Fig. 17 (ref. 42)).
Fig. 9 Phonon dispersion (PD) for group Pc1 calculated with increasing fineness of k-grid: (a) k = 0.04 A−1, (b) k = 0.02 A−1. |
Table 6 shows the frequencies and respective IR and Raman active modes for the lower symmetry super-lattices determined by these PD calculations. The majority of frequencies that correspond to symmetry P6/mmm are reproduced. Remarkably, additional frequencies between 230 cm−1 and 240 cm−1 occur. These frequencies are similar to values determined by the experimental measurements noted above. Additional higher frequencies also occur and appear similar to experimentally observed frequencies.
Frequency (cm−1) | Group | |||||||
---|---|---|---|---|---|---|---|---|
P3c1 | Pc1 | Pc2 | P63mc | |||||
158b | 165b | 188b | 186b | |||||
IR | R | IR | R | IR | R | IR | R | |
a These values display strong k-grid dependence with minor differences attributable to lattice parameter geometry optimization. (dd) indicates a doubly degenerate mode. b Value is the space group number. | ||||||||
233.4 (dd) | Y | Y | Y | N | N | Y | N | Y |
237.4 (dd) | Y | Y | N | Y | Y | Y | N | Y |
336.4 | N | N | N | N | N | N | N | N |
343.0 (dd) | Y | Y | Y | N | Y | Y | Y | Y |
395.4 | N | N | N | N | N | N | N | N |
415.9 | Y | Y | Y | N | Y | N | Y | Y |
534.9 (dd) | Y | Y | Y | N | Y | Y | Y | Y |
426.1–580.8a (dd) | Y | Y | N | Y | Y | Y | N | Y |
657.0 | N | N | N | N | N | N | Y | Y |
712.4 | Y | Y | N | Y | Y | N | N | N |
To place this study into perspective, we summarize below prior publications on Raman and IR spectroscopy of MgB2, in conjunction with prior modelling of MgB2 phonon behaviour. A consistent interpretation of these data through model and experiment is possible with an evaluation of symmetry conditions for the MgB2 structure.
While most Raman experiments detect the frequency ∼75 meV (= 604.9 cm−1), which corresponds to the E2g mode, measurements also identify peaks with magnitude and temperature dependence that appear closely related to the pair breaking energy or superconducting gap.1,6,9,10,24,55 For example, upon cooling of an MgB2 sample below 50 K, Kunc et al.1 have observed a build-up of additional scattering intensity in the range 50–300 cm−1 (Fig. 8(a) of ref. 1). This additional intensity can be modeled by two Gaussian peaks located at 128 and 226 cm−1, respectively, in the 2 K spectrum.1 These peaks are observed only for parallel polarizations of the incident and scattered light. With increasing temperature, these peaks vanish at a temperature between 74 K and 100 K; values that are well above Tc.1
Additional peaks are also observed in other Raman studies.5,9,11 These additional peaks are not attributable to the E2g mode nor to the pair breaking energy mentioned above and have not received much attention in the literature. These additional peaks are of weak magnitude and do not appear to show strong electron–phonon coupling. In addition, the conditions for detection of these weak peaks are limited, easy to confuse and readily attributed to an artifact. For example, Hlinka et al.5 have observed humps in Raman spectra of large crystals from thick films at about 300 cm−1, 400 cm−1, 600 cm−1, 750 cm−1 and 830 cm−1. These values correspond approximately to the reported peaks for PDOS computed in this study and documented above. Rafailov et al.11 investigated the Raman spectra of MgB2 and systematically compared the data with potential impurities. In their work,11 peaks at 250 cm−1, 600 cm−1 and 750 cm−1 are assigned to MgB2. However, in the same study, peaks at 255 cm−1, 517 cm−1, 1370 cm−1 and 1590 cm−1, and another set at 380 cm−1, 560 cm−1 and 960 cm−1 are measured as the light and dark areas (which we interpret to represent Mg and MgO), respectively, of an unreacted Mg stripe.11
According to Parlinski,36 far from the Γ-point, the acoustic PD relations reflect the vibrations of the Mg atoms in MgB2. The optic branches describe vibrations of the graphite-like boron network. Moreover, the branches of symmetry A2u and B1g at the Γ-point correspond to the out-of-plane boron vibrations, while four remaining curves with symmetry E1u and E2g at the Γ-point determine the in-plane boron motion. The out-of-plane and in-plane polarizations are approximately along the same branches all over the Brillouin zone.36
For reference, Fig. 10 shows the directions of vibration for boron atoms in the MgB2 structure defined by the E2g mode. A dynamic description of vibration modes for MgB2 is given by Yildirim.53
Fig. 10 Schematic showing directions of E2g vibrations (arrows) for boron atoms in MgB2. The dotted circles represent locations of Mg atoms above and below the boron plane. |
A random orientation of large grained, polycrystalline, materials and larger crystal faces of Sample A likely contribute to ready detection of additional peaks compared with Sample B albeit additional peaks are identified. The presence of extra peaks in Raman spectra of MgB2 is not exclusive to this work alone (see Section 4.1.1). In addition, several peaks appear to be both Raman and IR active. This outcome indicates not only that the symmetry is reduced with respect to P6/mmm, but also that the reduced symmetry may be non-centrosymmetric. For centrosymmetric materials, the rule of mutual exclusion for Raman and IR spectra requires a peak to be either Raman or IR active.49–51 However, this need not apply for non-centrosymmetric materials.
Earlier calculations of MgB2 vibrational behaviour26,54 demonstrate use of the frozen phonon approach. However, as mentioned by Kunc et al.,1 the frozen-in displacements required to perform these calculations lower the rotational symmetry of the system from D6h (or P6/mmm) to D3h, among other possibilities. Our calculations with reduced symmetry soften the constraints on Raman and IR activity without significant change to the frequency values as shown in Table 5.
Quantitative determination of specific factors such as anharmonicity,1,2,18,22,23,25,26 phonon anomalies,7,19,27 presence of impurities9,11,12,17 and multi-phonon contributions16,20,25 on the perturbation of first order vibrational features of MgB2 is outside the scope of this article. However, it is worth noting that the Raman spectra from these experiments show fluorescence in the 1000–4000 cm−1 range (see Fig. 3a). This fluorescence is likely to be related to the generation of second order vibrational features.58
Constructing a super-lattice to model structures with reduced symmetry appears to match earlier determinations of MgB2 phonon dispersions.21,32,34–36 However, the match between model and experiment is reasonable only if we consider that the multiplicity or near degeneracy of certain bands is partly hidden by experimental errors, or limited resolution, of the data. For instance, the E1u energy at the Γ point in Fig. 3 of the article by Shukla et al.32 appears to be the same energy as the top region of the highest energy acoustic mode at the M point. This study (Fig. 6 and 8) shows that a super-lattice with reduced symmetry reproduces the experimental data of Shukla et al.32 This alternative interpretation is informative, given that phonon dispersions in the work reported by Shukla et al.32 are constructed to fit experimental data from IXS2,18–20,32 and INS,21–23 respectively.
Symmetry conditions may also influence the accuracy of computational models for MgB2 spectra and the nature of Raman or IR activity. For example, experimental results and models more closely align when the 6-fold rotation axis is reduced to a 3-fold rotation axis, resulting in a doubling of the unit cell in the z-direction. This difference in symmetry description – which invokes a super-lattice in the z direction – predicts a higher number of Raman and IR active modes than with conventional six-fold symmetry. This outcome suggests that the Mg atoms have also lost 6-fold symmetry. The manifestation of this lower symmetry is an alternating Mg atom position towards and away from the plane containing boron atoms. Furthermore, the reduced symmetry super-lattice model predicts an additional frequency at about 235 cm−1. This frequency is observed in a number of experiments including this work as shown in Table 2.
On the other hand, the FD method, as exemplified by the extension of the LMTO method described by Savrasov,56 is an analytical version of a finite-difference approach within a super-lattice. When applied to the same problem, the results of both approaches, DFPT and FD, must be the same except for errors that may be introduced by taking finite differences.
As noted,39 phonon spectra in metallic compounds are very sensitive to details of the Fermi surface. To obtain reasonable spectra using the linear response method for metals, the k-point sampling must be dense and significantly higher density than the Fine mesh setting in CASTEP.39 Based on experience, we recommend setting the separation parameter on the k-points tab of the CASTEP electronic options dialog to less than 0.02 A−1. A clear sign of insufficient quality of k-point sampling is the presence of imaginary acoustic modes near the Γ-point.
In these calculations, the pseudo potentials mimic how valence electrons experience the screened core potential.59 In practice, agreement is expected between the integrated real and pseudo charge densities outside a chosen core radius for each valence state.60 The norm conservation criteria essentially ensures that the total charge of each pseudo wave function equals the charge of the all-electron wave function potential.59 For both DFPT and FD methods, changes in the E2g region of the PD are accompanied by changes in the ΓA, ΓM and ΓK directions. However, these changes are more pronounced in different phonon branches.
The PD calculated from the DFPT model, when using a sufficiently dense grid (k < 0.03 A−1) displays the typical “Mexican-hat” topology for the mode around E2g as shown in Fig. 6. According to Kunc et al.,1 this mode corresponds to vibrations in the boron plane. Such topology has been extensively discussed for systems displaying the Jahn–Teller (JT), or pseudo-Jahn–Teller effect61–64 and may be a signature of this effect in the MgB2 system. Moreover, calculations with LDA and GGA functionals result in identical PDs in DFPT, while LDA and GGA clearly differ in the FD approach. Apparently, the charge redistribution in the DFPT method, required with the charge renormalization for NCPs, is accounted for primarily by the “Mexican-hat” changes in the PDs. In comparison, charge re-distribution is not imposed to the same extent in the FD method and results in larger modifications of the PD in the acoustic band region.
Baron et al.19,66 identify the anomaly of the E2g mode as a Kohn anomaly,67 which originates from partial screening of lattice vibrations by the conduction electrons. This screening changes rapidly on certain surfaces in the space of phonon q-vectors and therefore, on these surfaces the frequencies vary abruptly with q.67
Baron et al.19 also report the presence of an anomalous optical mode in MgB2, similar in energy to that of the E2g mode, but with a different line width and symmetry. This mode is longitudinal along ΓA and not predicted by the theory used at the time. Remarkably, calculations of the PD shown in Fig. 9 (and listed in Table 6) for the double cell symmetry predict a flat branch at ∼650 cm−1, which is similar to values associated with the E2g mode.
This work highlights the dynamic nature of structural symmetry when phonons are included. Phonons are equivalent to dynamical perturbations of the lattice oscillating at the frequency ω, as pointed out by d'Astuto et al.2 As such, the phonon self-energy varies with time. To properly account for this, it is important to go beyond the adiabatic approximation to make use of time dependent perturbation theory.2 According to Calandra et al.,26 X-ray and Raman measurements are both explicable if dynamic effects beyond the adiabatic Born–Oppenheimer approximation as well as electron self-energy effects are included in the determination of phonon self-energy.
The use of reduced symmetry approximations as described in this study improves the description of phonon behaviour within the limitations of the adiabatic approximation. The potential JT-like PD that displays a “Mexican-hat” topology may be related to oscillatory electronic bands that are predicted to swing above and below the Fermi level as the lattice vibrates, with a corresponding charge population re-distribution when the band crosses the Fermi level.69–71
In this work, experimental data and computational models show that ∼230 cm−1 is an A-zone boundary frequency for the original P6/mmm unit cell (see Fig. 6 and 7). This frequency becomes part of the Γ-centre frequencies of the 2× super-lattice in the c-direction, following a procedure similar to that discussed by Kunc et al.1 and shown in Fig. 9. Inspection of the A-zone boundary for the 2× super-lattice also shows that a frequency ∼116 cm−1 has been created at the new A-zone boundary (see Fig. 9). Therefore, an extension of the super-lattice symmetry to a 4× cell in the same direction as the 2× super-lattice results in a frequency value at the Γ-centre zone of 116 cm−1. This analysis establishes a correlation between key phonon frequencies and a super-lattice with a commensurate modulation in the c-direction. Such a modulation is likely to be of a dynamic nature and involve electron transfers to bonds and spin polarizations that define the super-lattice.
A fourfold modulation in the c-direction (i.e. a 4× super-lattice) corresponds to approximately 4 × 3.5 A = 14 A. This value is approximately half the coherence length for MgB2 in the c-direction.72,73 If we consider that the phonon anomaly minimum also corresponds to a super-lattice modulation in the a–b plane, we obtain approximately 7 × 3.08 A = 21.56 A, for the period of modulation. A factor of 7 times arises because the bottom of the anomaly is located at ∼2 calculation points from the Γ-centre zone out of a total of 14 points that describe the direction to the K-zone boundary (see Fig. 6). A modulation in the a–b plane of ∼22 A is approximately half the coherence length of about 44 A for MgB2 in this direction.72,73
At the new zone boundary created by a super-lattice modulation in the a–b plane, extension of a line along the k-vector that corresponds to the minimum of the phonon anomaly determines the frequencies of the acoustic bands. This linear extension intersects the acoustic bands at frequencies ∼50 cm−1, ∼75 cm−1 and ∼120 cm−1, respectively (see Fig. 9). These three frequencies have been detected in Raman investigations of MgB2 with end values discussed in terms of pair breaking mechanisms and two superconducting gaps.15,55
A1g (= B1g) = E2g + 116 cm−1 |
where B1g and A1g are the highest frequencies in Tables 5 and 6, (B1g and A1g ∼ 710–715 cm−1) and E2g is ∼600 cm−1, respectively. B1g and A1g may convert with each other via accompanying charge movements that transforms group symmetry from P6/mmm to the lower symmetry double super-lattice. This equivalence provides not only a balance of energy, but an approximate mechanism where movements confined to the a–b plane may convert into movement in the z-direction leading to the creation of a switchable (on–off) dynamic super-lattice with corresponding boundary energy gaps.
An atom with three degrees of freedom has kBT/2 thermal energy per degree of freedom.74,75 Since MgB2 has three atoms per unit cell, the thermal energy per unit cell of MgB2 is 4.5kBT. At Tc, this thermal energy provides energy just above the superconducting gap 2Δ = 4kBTc, and thus, suggests that excitation of phonon modes that are separated by a gap energy is likely to be a significant determinant of superconductivity.
Fig. 11 shows a plot of the calculated frequencies for the 2× super-lattice using spectral values from Table 6, assuming integer multiples of the energy equivalent to the vibration mode of the lower symmetry structure noted in 4.7 above (i.e. ∼116 cm−1). These ab initio calculated frequencies define a linear trend based on the major acoustic mode for this super-lattice symmetry with other nearly parallel, slightly offset values. This trend suggests that the frequencies of the MgB2 structure at the Γ-point have some similarity to those of an ‘harmonic oscillator’, where the allowed frequencies are expressed as integer multiples of a basic value.76–79
Fig. 11 Ab initio calculated frequencies at the Γ centre point derived from Table 6 compared as integer multiples of an energy value (∼116 cm−1) that correspond to the major acoustic energies for an MgB2 super-lattice. |
Coherent phonon relaxation has been investigated in semiconductors50,80 and is the basis for coherent phonon spectroscopy.81 Coherent phonon relaxation is described in terms of phonon decay time and dephasing.50,80,81 Decay times are typically longer at lower temperatures and dephasing may induce anharmonicity. Primary mechanisms in coherent phonon relaxation are up-conversion of modes by thermal phonons as well as down-conversion by decay of optical phonons into acoustic phonons.50,80,81 In this latter case, acoustic phonons carry half the energy of the predecessor optical phonon but with opposing k vectors, thus conserving energy and momentum while avoiding incoherent scattering.
The linear integer proportionality of phonon frequencies at the Γ-point for MgB2 noted above suggests that conversion of modes – where energy and momentum as well as phase coherence are conserved – is inferred from our combined experimental data and DFT calculations. For example, if the temperature cannot excite up-conversion from a particular energy level to the next higher energy mode, all remaining relaxation options are decay modes with phase coherent mechanisms.
This mechanism is exemplified by the available conversion processes from the important E2g mode (∼581 cm−1). For example, when temperature remains above Tc, excitation into the 712 cm−1 mode is enabled by thermal phonons. Down-conversion from this higher level leads to scattering and resistive losses because decay paths with exclusively coherent options cannot be established. Once the temperature cannot excite up-conversion to the 712 cm−1 mode, the only remaining options for relaxation are decay modes within major phase coherent mechanisms.
Decay from the E2g mode at ∼581 cm−1via two acoustic modes at ∼116 cm−1 (with opposing momentum) lands at an energy level of 351 cm−1 which is observed in Raman spectra of MgB2, (as shown in Table 2), and is also calculated for the 2× super-lattice (as shown in Table 6). The energy at 351 cm−1 belongs to the set of multiple integer values for MgB2. Such a mechanism is not accessible for decay from the 712 cm−1 mode, since two acoustic modes at ∼116 cm−1 do not land within available integer values. It is worth noting that the 351 cm−1 mode energy is approximately equivalent to half the B1g and A1g energy levels (at ∼712 cm−1), which may also be important for energy conservation.
PDOS calculations are insensitive to subtle changes in the PD produced by both commensurate and incommensurate super-lattice modulations. Similarly, frozen phonon approaches, which are limited to accurate determination of vibration frequencies at the Γ point, also mask crucial information at super-lattice positions. Computationally, the use of UPs appears to mask fine-scale charge re-distributions affecting the PD. Such charge re-distribution is associated with fluctuations of the nearly flat bands running parallel to the Fermi energy.69–71
DFT calculations of PDs and vibrational frequencies associated with a double cell in the c-direction explain the observed experimental data not only in this study but from several others which use different analytical techniques. PDs calculated with a 2× super-lattice produce a better overall shape and match to PDs determined from IXS and INS experimental data. Use of super-lattice symmetries for MgB2 is a better-modelled approximation to dynamic modulations introduced by phonons.
Energy conservation through conversion of phonon energies by coherent relaxation may be manifest in the set of multiple integer frequencies reflected in the Raman and IR spectra of MgB2 and may be a potential contributing mechanism for superconductivity in this material. This proposition is modelled using DFT and establishes a potential structural foundation for several key superconducting parameters. Extension of these concepts to other superconducting systems is currently in progress.
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