An Inelastic X-Ray investigation of the Ferroelectric Transition in SnTe

TL;DR

This study uses inelastic X-ray scattering to demonstrate that SnTe undergoes a ferroelectric transition characterized by a softening and subsequent increase in energy of a polar phonon mode below 75 K, despite its metallic nature.

Contribution

It provides direct evidence of ferroelectric displacement in metallic SnTe and shows that its transition resembles that of insulators, highlighting the role of phonon interactions over electron effects.

Findings

The transverse-optic phonon softens to near zero at T_C=75 K.

The polar mode energy increases as temperature decreases below T_C.

The transition is a nearly ideal continuous phase transition.

Abstract

We report that the lowest energy transverse-optic phonon in metallic SnTe softens to near zero energy at the structural transition at $T_C=75 \text{~K}$ and importantly show that the energy of this mode below $T_C$ increases as the temperature decreases. Since the mode is a polar displacement this proves unambiguously that SnTe undergoes a ferroelectric displacement below $T_C$. Concentration gradients and imperfect stoichiometry in large crystals may explain why this was not seen in previous inelastic neutron scattering studies. Despite SnTe being metallic we find that the ferroelectric transition is similar to that in ferroelectric insulators, unmodified by the presence of conduction electrons: we find that (i) the damping of the polar mode is dominated by coupling to acoustic phonons rather than electron-phonon coupling (ii) the transition is almost an ideal continuous transition (iii) comparison with density functional calculations identifies the importance of dipolar-dipolar screening for understanding this behaviour.

Equations

$$\Delta {\omega }_{\text{TO}}=K\frac{n[\beta {\omega }_A]-n[\beta ({\omega }_{\text{TO}}(0)+{\omega }_A)]}{{\omega }_{\text{TO}}(0)({\omega }_{\text{TO}}(0)+{\omega }_A)}$$

$$K=\frac{16}{27\pi }\frac{Q^2{c}^2}{{\rho }^3{a}^7}\frac{\hslash ({\mathbf{q}}_{c}a{)}^3}{v_s(v_s-v_{\text{TO}})}.$$

$$K\sim 0.08Q^2{\text{ [meV]}}^3.$$

Full text

SnTe was originally studied in the context of lattice vibrations in diatomic lattices Pawley et al. (1966). There has been a recent resurgence of interest following its identification as a crystalline topological insulator, which is intimately related to its room temperature structure (the $fcc$ rocksalt structure) Hsieh et al. (2012); Tanaka et al. (2012); Li et al. (2016). Stoichiometric SnTe is expected to be semiconducting with the minimum gap in the bandstructure of approximately 0.1 eV at the $L$ point in the Brillouin zone  Tung and Cohen (1969); Rabii (1969); Littlewood et al. (2010). Perfectly stoichiometric SnTe has, however, never been grown. Instead the crystals are always Te rich Brebrick (1963); Savage et al. (1972), with the extra Te being accommodated in the lattice by Sn vacancies. The vacancies lead to a high free carrier concentration of holes, $n_{h}$, that do not freeze out at low temperature. Moreover, the material undergoes a phase transition to a rhombohedral structure upon cooling. Such a structural transition would then strongly affect some of its topologically protected states Hsieh et al. (2012), as has been reported in the Pb1-xSnxSe class of crystalline topological insulators Okada et al. (2013). The exact transition temperature, $T_{c}$, is dependent on the value of $n_{h}$ Kobayashi et al. (1976).

While the transition is predicted to be a displacive ferroelectric transition Littlewood (1980); Littlewood. (1980); Salje et al. (2010); Rabe and Joannopoulos (1985), no ferroelectric response has previously been seen in bulk samples due to screening by the free carriers. However a ferroelectric like response has been reported recently in ultrathin films Chang et al. (2016). Whether this response co-exists with metallic conductivity is not clear. A shift of much of the electronic density of states (eDOS) within 2 eV below $E_{f}$ was seen with ARPES Littlewood et al. (2010), suggesting an electronic mechanism. It is unclear whether the polarisation is the primary order parameter or a secondary order parameter as in some manganites Efremov et al. (2004). For an insulator the divergence (or not) of the dielectric constant at $T_{C}$ resolves this issue. For metallic SnTe direct evidence establishing that polarisation is the primary order parameter is however missing.

The non-stoichiometry means that SnTe is not insulating but a degenerate semiconductor (with a finite resistivity at low temperature). Ferroelectricity and metallic conductivity are different macroscopic responses, with the former involving the separation of localized charges, while the latter is a property of delocalized electrons. Since local charges are strongly screened by conduction electrons, the two properties might be considered to be mutually exclusive. Theoretically, they may however co-exist Anderson and Blount (1965), and such a coexistence was found in recent experimental studies of BaTiO3-δ Kolodiazhnyi et al. (2010) and LiOsO3 Shi et al. (2013). Co-existence is possible for electron concentrations up to a critical value at which the Thomas-Fermi screening length falls below a critical correlation length for ferroelectricity Kolodiazhnyi et al. (2010).

Generally, ferroelectric displacive transitions are expected to be accompanied by changes in the dynamics of the lattice with the energy of a soft phonon mode at some high symmetry $\bf q$-points (either at the zone centre or edge) decreasing on cooling to a minimum at $T_{c}$, only reaching zero for a continuous transition and then rising Shirane (1974); Cochran (1960). A typical example is the $1^{st}$ order structural transition from cubic to a ferroelectric tetragonal phase in PbTiO3 Shirane et al. (1970); Kempa et al. (2006); Tomeno et al. (2006), which is driven by a soft transverse optic mode at the Brillouin zone centre. Empirically ferroelectric transitions are almost all first order. The reason for this is not clear although the long range nature of the dipole-dipole interactions is sometimes credited Lines and Glass (1977). The coupling between the polarisation and strain further favours a first-order instability Chandra and Littlewood (2007). In contrast for SnTe we report that the phonon frequency approaches zero at the structural transition consistent with a continuous or very weakly first order transition.

There are many examples of nearly continuous structural distortions in metals such as the cubic to tetragonal transition in V3Si and Nb3Si Testardi (1975) driven by a soft acoustic mode towards the zone edge. However these martensitic transitions are not ferroelectric. Our identification that a zone centre transverse optic phonon softens and recovers going through $T_{c}$ in SnTe indicates its transition is a ferroelectric transition like that in PbTiO3.

Previous investigations of the phonon dispersion curves in SnTe carried out by Pawley $et\;al.$ Pawley et al. (1966) and more recently Li $et\;al.$ Li et al. (2014) using inelastic neutron scattering found significant softening of the transverse optic phonon energy towards the Brillouin zone centre ($\Gamma$ point). Similar softening has been reported in the binary material Pb1-xSnxTe Dolling and Buyers (1973). Although the softening was strongly temperature dependent, phonon energies never softened close to zero even at the lowest measured temperatures. Therefore these authors suggested SnTe and Pb1-xSnxTe approached ferroelectric transitions without ever passing through them. However indications of a structural distortion have been reported in other experiments such as powder X-ray diffraction where peak splitting was seen consistent with the cubic phase being distorted along (1,1,1) directions to a rhombohedral shear angle of $\delta\alpha\sim 0.115^{\circ}$ Muldawer (1973). Other signatures of the transition are a change of Bragg reflection intensity in neutron scattering Iizumi et al. (1975) consistent with a relative shift of the two $fcc$ sublattices of $\tau\sim 0.007$ ($\sim 4$ pm), and changes in the Raman spectrum Brillson et al. (1974) and heat capacity Hatta and Kobayashi (1977). A cusp-like anomaly in the electrical resistivity at $T_{c}$ Kobayashi et al. (1975); Grassie et al. (1979) has been attributed to the presence of a soft phonon mode. The samples previously studied with inelastic neutron scattering most likely had a value of $n_{h}$ that was sufficiently high for $T_{c}$ to have been suppressed. A clear proof that the observed structural transition is a ferroelectric displacement is then lacking. To determine whether SnTe is indeed ferroelectric, we performed inelastic x-ray measurements on a small, close to stoichiometric, single crystal.

The single crystal was grown from equal molar weights of high purity elements Sn (99.9999$\%$) and Te (99.9999 $\%$) wrapped in Mo foil (99.95$\%$), sealed in an evacuated silica ampoule and heated to $850^{\circ}\textrm{C}$, well above the melting temperature Brebrick (1963). The mixture was slowly cooled to $760^{\circ}\textrm{C}$ and the ampoule then quenched into water at room temperature. The structural transition at $T_{c}=$75 K was identified in the extracted crystal from a clear change in the slope of the electrical resistivity (see supplementary material). A value of $n_{h}=3.23\pm 0.05\times 10^{20}\textrm{cm}^{-3}$ was deduced from a measurement of the Hall resistivity at 2 K (see supplementary material). The value of $T_{c}$ = 75 K is in good agreement with previous measurements for similar $n_{h}$ Kobayashi et al. (1976).

Following an initial examination with diffuse scattering (supplementary material), the single crystal was studied by inelastic X-ray scattering with the ID28 instrument at the ESRF (Grenoble). A backscattered silicon (9, 9, 9) monochromator was used. The scattered photons were analysed with 9 single crystalline spherical silicon analysers mounted in a pseudo Rowland circle geometry. Energy scans were performed at constant $\bf Q$-vector by varying the temperature of the monochromator covering the range -19.82meV to 19.82meV in 0.68 meV steps centred at the analyser energy of 17.794 keV. The penetration depth of the incident X-rays was $45\mu$m, ensuring reported results are bulk properties. The energy resolution had a Lorentzian lineshape with a 3meV FWHM. Measurements were made over different zones to distinguish the different phonons. The experimental spectra are well described by the instrument resolution convolved with antisymmetrised pairs of Lorentzians for each mode, respecting detailed balance C.Stock et al. (2004), plus a narrow fixed width Lorentzian at the elastic position (see FIG 2 (b) for an example).

Electronic structure calculations were performed for the stoichiometric compound, using scalar-relativistic density functional theory (DFT) as implemented in the VASP package, Kresse and Furthmüller (1996); Kresse and Joubert (1999); John P. Perdew, Burke and Ernzerhof (1996), a plane wave basis with cutoff energy $E_{c}=200$ eV and k-point sampling with a linear density of $30/\text{\AA}^{-1}$ (grid size $(8,8,8)$). The energy difference between the cubic and rhombohedral structures converged to within 0.05 meV/formula. We found the ground state energy for the rhombohedral structure to be about 0.2 meV/formula lower than for the cubic structure, with an optimum rhombohedral angular distortion of $\delta\alpha\approx 0.2^{\circ}$ and a ferroelectric displacement of $\tau\approx 0.009$. Ground state phonon dispersion calculations used the finite displacement method Alfé (2009) in 128-atom supercells and confirmed the dynamical stability of the rhombohedral phase and the instability of the cubic phase (supplementary material). The self-consistent ab-initio lattice dynamics method (SCAILD) was then used to obtain phonon dispersion curves at finite temperatures for the cubic phase P Souvatzis and Rudin (2008). The calculated phonon dispersion for the two structures (rhombohedral at T$=0$ and cubic at T$=300~{}$K) are very similar over most of the Brillouin zone. The main differences are close to the zone centre where there is a sharp reduction of the energy of the LO ($\Delta_{1}(O)$) mode for the rhombohedral structure and a more modest reduction of the TO ($\Delta_{5}(O)\rightarrow\Delta_{2}(O)+\Delta_{5}(O))$ mode (figure 1 and supplementary material). The former may be a consequence of modulations of the polarisation present in the rhombohedral structure more effectively screening the LO phonon.

In Figure 1 the cubic phase calculation is compared with the measured phonon dispersion curves at 300 K (momentum vectors are given with respect to the conventional cubic unit cell all throughout the paper). No experimental points for the $\Sigma_{4}(O)$ branch could be accurately determined. Our measurements agree with previous inelastic neutron scattering measurements above $T_{c}$ Cowley et al. (1969). The sharp decrease in the LO phonon energy towards $\Gamma$ and dip along the (1,1,0) direction in the measurements relative to the calculation have previously been attributed to the macroscopic electric field associated with the phonon coupling to free charge carriers Cowley and Dolling (1965); Cowley et al. (1969). The LO phonon dips over a range of $\zeta<0.2~{}a^{*}$ around $\Gamma$ (figure 1). The reciprocal-space width over which the the TO phonon shows a temperature dependence ($\zeta>0.2~{}a^{*}$, figure 2 a) is larger than this, supporting the hypothesis that the Thomas Fermi-screening length exceeds the ferroelectric correlation length. However, the LO dispersion measured with neutrons by Pawley Pawley et al. (1966) is indistinguishable from our measurements, although a larger range for the dip, reflecting a shorter Thomas-Fermi screening length, might have been expected from the absence of ferroelectric order in their crystal. This is reconciled by noting that the polarisation may provide the dominant screening mechanism, explaining the dip in the LO mode Cowley et al. (1969).

Phonon dispersion curves at $T_{c}$ = 75 K are shown in the supplementary material. There is no measurable change of the experimentally determined elastic constants between 300 K and 75 K consistent with previous measurements Beattie (1969); Salje et al. (2010). The only observable temperature dependence is for the TO mode approaching $\Gamma$ and we focus on this in the following.

The measured energies of the transverse optic (TO) phonon at 300 K and at $T_{c}$ = 75 K are shown in Figure 2 (a). A softening of the phonon energy towards the Brillouin zone centre, $\Gamma$, at low temperature is clearly seen. Measured spectra at ${\bf Q}=(3,3,0.2)$ are shown in Figure 2 (b) at 300 K and 75 K; the 3 peaks correspond to phonon creation (negative energy), phonon annihilation (positive energy) and a small elastic contribution at zero energy. The resolution function makes very low energy phonons difficult to distinguish from the elastic line close to the zone centre where the Bragg condition is met. Thus, in Fig 2 (a) the 75 K data is missing a point at ${\bf Q}=(3,3,0.95)$. Further temperature points were measured for the TO phonon along the (1,1,0) direction. This confirms the softening on cooling to $T_{c}$ (Figure 2 (c)), while Figure 2 (d) shows that on further cooling below $T_{c}$ to 50 K and then to 25 K the phonon energy recovers. Individual scans at ${\bf Q}=(0.1,0.1,5)$ are shown in Figures 3 (a-c), along with calculated fits unconvoluted from the instrument resolution to judge their reliability. For comparison the measured spectra for the transverse acoustic (TA) phonon at 300 K and $T_{c}$ are shown in Figures 3 (d) and (e) at ${\bf Q}=(1.8,1.8,4.2)$, the closest point to $\Gamma$ that could be measured accurately, showing that this mode does not change with temperature.

The values of phonon energies at $\Gamma$ were determined by linearly extrapolating the phonon energy squared $E({\bf q})^{2}$ plotted against ${\bf q}^{2}$ to ${\bf q}^{2}\rightarrow 0$ (supplementary material), where q is the reduced momentum. Figure 4 (a) shows that $(E_{\text{TO}}(\Gamma))^{2}$ decreases linearly with temperature to almost zero at $T_{c}$ and increases again above $T_{c}$. In the Landau theory of ferroelectrics the ratio of the slopes $d((E_{\text{TO}}(\Gamma))^{2})/dT$ below $T_{c}$ to above $T_{c}$ should be -2. The electrical resistivity was previously found to be adequately described with a simple model calculation based on this ratio for a temperature range $-0.35<(T-T_{c})/T_{c}<1.4$ Kobayashi et al. (1975). The solid line in the figure is a fit assuming the Landau theory ratio and including a saturation of $E({\bf q})^{2}$ towards a fixed value at high temperature reflecting the much larger temperature range $-0.35<(T-T_{c})/T_{c}<4$ spanned by our measurements.

The continuous ‘$2^{nd}$ order’ nature of the transition, previously suggested by heat capacity measurements Hatta and Kobayashi (1977), is reflected in Figure 4 (a) by the nearly complete softening to zero energy of $E_{\text{TO}}(\Gamma)$ at $T_{c}$. Such a complete softening of the phonon energy is highly unusual for ferroelectric transitions in insulators.

The coupling of LO and TO optic modes has been argued to be responsible for a large phonon non-harmonicity in PbTe and SnTe Delaire et al. (2011); Li et al. (2014) that may explain the high figures of merit of these materials for thermoelectric applications. We found no temperature dependence or abnormal damping of the LO phonon to support this.

We now discuss the line-width of the TO phonon. The line-width as a function of $\bf q$ increases sharply towards $\Gamma$ above $T_{c}$ but is small and constant in $\bf q$ below $T_{c}$ (supplementary material). Figure 4(b) shows the line-width as a function of temperature at ${\bf Q}=(0.1,0.1,5)$ and ${\bf Q}=(0.125,0.125,5)$. The TO line-width is enhanced at the zone centre above $T_{c}$ and this enhancement is suppressed below $T_{c}$. Figure 4(c) shows the line-width of the TA phonon at ${\bf Q}=(1.8,1.8,4.2)$ for comparison which is much smaller and shows no significant temperature dependence.

We find that a very simple model for anharmonicity based on the phonon interaction $TO(0)+TA({\bf q})\leftrightarrow TO({\bf q})$ or $TO(0)+LA({\bf q})\leftrightarrow TO({\bf q})$ Dvořák (1967) can explain our measurements. The calculated curve in figure 4 (b) is determined from $E_{\text{TO}}(\Gamma)$ taken from Figure 4(a) (solid line) and a fixed intercept with an acoustic phonon branch at 4 meV (see supplementary material). There is only one further parameter in the calculation that fixes the overall amplitude of the damping. This parameter is proportional to the ratio $Q$ of the strain $\epsilon$ induced by the displacement to the squared ferroelectric displacement $\tau^{2}$. The value required to fit the data gives $Q=\epsilon/\tau^{2}=17$. This value of $Q$ agrees well with an estimate from our DFT calculation $Q_{\text{DFT}}=13.4$ which also reproduces the observed magnitudes of $\alpha$ and $\tau$. This strongly suggest that phonon anharmonicity from coupling to acoustic phonons indeed explains the measured TO phonon line-width.

The electron TO-phonon interaction has been considered to drive the displacement transition Kobayashi et al. (1976). Although this interaction and its modification with doping may play a major role in determining the TO phonon energy our analysis shows that it contributes only indirectly to the phonon linewidth via the phonon energy and the strain-displacement coupling. A direct electron-phonon contribution to the line width from the conduction electrons is not needed, however can not be ruled out. We also note that there is no significant change in the Hall resistivity between 150 K and 2 K (supplementary material), ruling out changes in the number of conduction electrons as the cause of the drop in the line width below $T_{c}$.

In summary, we have shown for the first time that the structural transition in a metallic sample of SnTe is a ferroelectric displacement transition with the TO phonon hardening below $T_{c}$. Such a recovery of the phonon energy below $T_{c}$ has never been reported before, despite being highly sought after, and confirms that polarisation is the primary order parameter. We found that the damping of the TO phonons close to the zone centre can be explained by a conventional coupling of displacement and strain. Such a coupling acts to make the transition first-order. The continuous (or only weakly first order) nature of the transition then requires that the dipole mode-mode coupling term in a Landau expansion of the free energy (the $4^{th}$ order term in powers of the polarisation) is large in the absence of strain. The marked difference of the measured LO phonon energy from that calculated with DFT suggests that a strong many-body dipole-dipole screening may be present in the parent insulating material. Identical LO phonon energies across samples where the structural transition is present or absent also suggest that $T_{c}$ is not suppressed by screening of the dipole-dipole interaction through added charge carriers. Another mechanism for the suppression is provided in ref. Kobayashi et al. (1976) where it is suggested that it is due to the removal of valence electrons rather than screening from conduction electrons. Our results support such a diminished role for the conduction electrons and indicate that their contribution to characteristics such as the soft-mode phonon line-width is minor.