Parameters of alloys necessary to calculate the anharmonic contribution to the phonon frequency change
1. Introduction
The cubic IIVI systems HgTe, CdTe and ZnTe are extremely interesting technologically materials with many applications such as infrared as well as quantum electronics devices [21, 29, 5, 44; 26]. The phonon frequencies of these alloys belong to the farinfrared region and investigation of their phonon spectra was one of more important problem in the infrared spectroscopy during 70th years. The temperature dependence of the phonon HgTemode frequencies in binary HgTe, ternary HgCdTe (MCT) and HgZnTe (MZT) materials has been the subject of an intense debate in the last three decades [16, 14, 2,1, 29, 4; Kozyrev et al., 1996; 31, 12, 35, 28], due to contradictory results regarding in particular the abnormal temperature shift of the HgTelike TOphonon frequency. The latter is opposite to the normal phonon frequency temperature shift of many alkali compounds as well as most of semiconductors. In fact, the HgTelike TOphonon frequency increases when temperature increases while the normal temperature shift associated with a crystal lattice expansion, has to be opposite: the frequency decreases when temperature is raised.
The different behaviour has been qualitatively explained by an electronphonon (
The
On the other hand, reverse effects, i.e., how light electrons affect the phonon spectrum are much less known. Recently, we have shown [35] that the singularity in an electron energy spectrum induces a discontinuity in the phonon frequency temperature dependence of the Hg_{1−x} Cd_{x}Te alloys. That means the appearance of an unexpected effect of strong resonance influence on oscillations of heavy atoms by the electron subsystem. In our short communication [36] we called this effect as returnable ep interaction. It was shown that even though the returnable epcoupling has a nonpolarized character, yet a deformation mechanism takes place [35, 36, 19]. It is necessary to note that singularity in the electron spectrum that induced the resonance returnable
It is then important to test the occurrence of this or similar effects in other semiconductor compounds. Moreover, it seems that the abnormal temperature dependence of the HgTelike phonon modes can be solved in this framework, because the presence of a returnable ep interaction could explain the positive temperature shift of the optical phonon frequencies. On the other hand, the positive temperature shift of the phonon frequency is characteristic for the HgTelike modes in different Hgbased alloys such as the abovementioned HgCdTe, the HgCdSe [40, 39] and the HgZnTe as shown in the sequel. Data suggest that a spinorbit relativistic contribution, larger for heavy atoms, play an important role in this phenomenon because of its effect on the chemical bond [42,17, 30].
The aim of this contribution is to generalize experimental data on the temperature dependence of the TOphonon modes in the Hg_{1−x} Cd_{x}Te and Hg_{1−x} Zn_{x}Te alloys of different compositions and analyze the influence of the resonance returnable epinteraction in case when the temperatures are close to singular Dirac point (E_{g}≡ Γ_{6}− Γ_{8}=0) and far from this point for alloys where the Dirac point exists. Such analyses should be performed on a background of obligatory anharmonic contribution caused by the temperature extension of the crystal lattice. That analyses should be distributed on the alloy compositions where the Dirac point exists not, also.
The rich experimental reflectivity data in the far IR region of MCT and MZT alloys that were collected during 20022006 years for different compositions and in a wide temperature range using a synchrotron radiation source [34, 27, Cebulski et al., 2008; 35, 36, 28] allow unique opportunity for such investigation.
2. Experiment
In order to investigate the temperature behaviour of the phonon modes for HgZnTe and HgCdTe alloys, several optical reflectivity measurements were performed in the farIR region at the DAΦNElight laboratory at Laboratori Nazionale di Frascati (Italy) using a synchrotron radiation source (details on the experimental setup are available in work(Cestelli Guidi et al., 2005)). A BRUKER Equinox 55 FTIR interferometer modified to collect spectra in vacuum, was used. As IR sources both the synchrotron radiation from the DAΦNE storage ring and a mercury lamp were used. Measurements were performed from 20 to 300 K and in the wavenumber range 50600 cm^{−1}. In order to provide the spectral resolution of 1 cm^{−1} (2 cm^{−1} in some cases), we typically collected 200 scans within 600 s of acquisition time with a bolometer cooled to 4.2 K. The reflectivity was measured using as a reference a gold film evaporated onto the surface of the investigated samples. This method enabled us to measure the reflectivity coefficient R(ω, T) with an accuracy of 0.2%. The
Hg_{1−x} Cdx Te crystals were grown at the Institute of Physics of the Polish Academy of Sciences in Warsaw (Poland) while the Hg_{1−x} Zn_{x} Te ones at the CNRSGroupe dEtude de la Matire Condense (Meudon, France). The reflectivity curves R(ω, T) for Hg_{0.90}Zn_{0.10}Te in the frequency range from 80 cm^{−1} to 220 cm^{1} and in the temperature range 30300 K are shown in Fig. 1.
Data show that the main phonon band consists of two subbands: a HgTelike band in the range 118135 cm^{−1} and a ZnTelike band in the range of 160180 cm^{−1}. Both of them are characterized by a fine structure. A nonmonotonic temperature dependence of the reflectivity maxima can be also recognized. Similar R(ω, T) curves are showed in Fig.2 for the Hg_{0.763}Zn_{0.237}Te. The maxima on reflectivity curves in Fig. 2 are shifted with increase of temperature towards lower phonon frequencies monotonically.
To recognize the real frequency positions of a phonon mode, it is however necessary to calculate for each obtained experimental curve R(ω, T) the imaginary part of the dielectric function Im[ϵ(ω, T)] as function of frequency and temperature. That ones were calculated from the reflectivity spectra shown in Fig. 1 and 2 by means of the KramersKronig (KK) procedure. This procedure is described in details in the work of [6]; was applied to experimental results presented in several papers, for example [35, 28, 25]. An estimated uncertainty of 1.5% takes place at calculation the Im[ϵ(ω, T)] curves for all experimental data. The Im[ϵ(ω, T)] curves at different temperatures are shown in Fig. 3 a,b for the Hg_{0.90} Zn_{0.10} Te sample as well as in Fig.4 a,b for the Hg_{0.763} Zn_{0.237} Te sample. In Fig. 3 the Im[ϵ(ω, T)]curves are presented separately for HgTeband (Fig. 3a) and for ZnTeband (Fig.3b).
It is necessary to underline here that the maximum of the HgTelike subband (Fig. 3a) shifts towards higher frequencies when the temperature increases from 30 K to 80 K while at temperature higher than 85 K the maximum shifts to lower frequencies. A nonmonotonic temperature dependence of the ZnTelike subband (Fig. 3b) with maximum frequency position near 85 K is also observed.
The frequency positions of HgTelike and ZnTelike subband maxima determined from the Im[ϵ(ω, T)] curves at different temperatures in the range 30300 K are shown for the sample Hg_{0.90} Zn_{0.10} Te in Fig. 5a and 5b, respectively.
The KKtransformation was also performed for the R(ω, T) curves shown in Fig.2 (the Hg_{0.763} Zn_{0.237} Te sample). The Im[ϵ(ω, T)] curves at different temperatures for this sample are shown in Fig. 4, while the positions of the Im[ϵ(ω, T)]maxima are shown in Fig.6 a,b for the HgTelike mode and the ZnTelike mode, respectively.
Analogous investigations of the temperature dependence of the phonon frequencies for different modes were performed for another semiconductor alloy contained the mercury as one of component, namely: HgCdTe. The results were published earlier: [35, Sheregii et al. (2010), 28]. The measured curves of R(ω,T) for Hg_{0.885} Cd_{0.115} Te in the frequency region from 80 cm^{1} to 170 cm^{1} and the temperature interval 40 K – 300 K are shown in Fig. 7. From Fig. 7 it is clearly seen, that the main phonon band consists of two subbands: a HgTelike band in the range of 11813 cm^{1} and a CdTelike band in the range of 140160 cm^{1} both characterized by a fine structure well known for the alloy phonon spectra [43, Kozyrev and Vodopynov 1996, 25]. A nonmonotonic dependencies of the reflectivity maxima are seen too. The Im[ε(ω,Τ)] curves calculated from the reflectivity spectra are showed in Fig. 8 a,b. We have also to underline here that maximum of the HgTelike subband is shifted towards higher frequencies when the temperature increases from 170 K to 240 K while for temperature higher than 240 K the maximum is shifted to lower frequencies. The similar temperature behaviour demonstrates the CdTelike subband maximum.
The VerleurBarker model [43] with five structural cells together with the statistical approach developed recently [26] is applied to the Hg_{1−x} Zn_{x} Te solid solutions. According to this model each of the two subbands: HgTelike and ZnTelike in the case of the Hg_{1−x} Zn_{x} Te alloys, consists of not more the four modes due to the oscillations of the HgTe or ZnTe dipole pairs in each of the five tetrahedra T_{n}, where n is the number of Znatoms in the cell. Therefore, the maximum of each subbands can be associated to one of these four modes depending on the composition of alloy. In the case of the Hg_{0.90} Zn_{0.10} Te alloy we combine the T_{0}mode with the HgTelike subband and the T_{1}mode with the ZnTelike mode, similarly to the Hg_{0.85} Cd_{0.15} Te alloy. Regarding the Hg0.763 Zn0.237 Te alloy the maximum of the HgTelike subband is attributed to the T_{1}mode, while the maximum of the ZnTelike subband, to the T_{2}mode. A comparison of the temperature dependence of the TOmode frequencies showed in Fig.3 and 6 point out a monotonic behaviour of the curves of the Hg_{0.763}Zn_{0.237}Te sample (Fig.6 a,b), while discontinuities occur in the curves of the semimetallic composition Hg_{0.9} Zn_{0.1} Te (Fig.4 and 5) with a positive temperature shift of the HgTelike mode frequency and a negative temperature shift, for both compositions, of the ZnTelike mode. In the sample Hg_{0.763} Zn_{0.237} Te a similar temperature dependence of the HgTelike and ZnTelike TOmodes as in the sample Hg_{0.80} Cd_{0.20} Te are observed the HgTelike and CdTelike TOmodes as shown in Fig. 10 a and b, respectively. For the sample Hg_{0.763} Zn_{0.237} Te, the frequency of the HgTelike mode is practically independent of the temperature.
In Fig. 9 are shown the frequency positions of the HgTelike and CdTelike subband maxima on the Im[ε(ω,Τ)] curves for sample Hg_{0.885}Cd_{0.115}Te. It is seen from Fig. 9 that discontinuity is taken place precisely at 245 K for both HgTeand CdTe modes. Generally, for CdTemode is observed negative temperature shift of the phonon frequency, while for HgTemode is positive one.
It is interesting to compare the temperature dependences of the phonon modes for another alloy of HgCdTe. As was shown in Fig. 6b for the sample Hg_{0.763} Zn_{0.237} Te, the frequency of the HgTelike mode is practically independent of the temperature. Similar behaviour takes place for CdTemode in alloy Hg_{0.80}Cd_{0.20}Te as it is seen in Fig. 10 where are shown the temperature dependences for the phonon mode frequencies for this alloy.
So, it is possible to generalise the obtain experimental results on temperature behaviour of the phonon spectra for four alloys contained the mercury – Hg_{1x} Zn_{x} Te and Hg_{1x} Cd_{x} Te. If composition x is near the value where the particularity in energy structure takes place, namely the Dirac point (E_{g} ≡ Γ_{6} − Γ_{8}=0), for example in the case of the Hg_{1x} Cd_{x} Te alloys that is x=0.1 – 0.17 and in the case of the Hg_{1x} Zn_{x} Te alloys it is x=0.06 – 0.11, then positive temperature shift is observed for the HgTemodes with discontinuity the temperature where the Dirac point takes place.





Hg_{0.85}Cd_{0.15}Te  HgTelike  121.8  3.7  1.9 
CdTelike  152.0  2.3  0.4  
Hg_{0.80}Cd_{0.20}Te  HgTelike  118.0  4  0.6 
CdTelike  152.4  2  0.4  
Hg_{0.90}Zn_{0.10}Te  HgTelike  119.2  3.6  1.04 
ZnTelike  171.2  0.26  1.87  
Hg_{0.763}Zn_{0.237}Te  HgTelike  126.03  0.001  0.005 
ZnTelike  177.0  10  2 





Hg_{0.85}Cd_{0.15}Te  8  5  6.49  6 
Hg_{0.80}Cd_{0.20}Te  8  5  6.49  1 
Hg_{0.90}Zn_{0.10}Te  7  3  6.63  6 
Hg_{0.763}Zn_{0.237}Te  7  3  6.65  1 
In the case of compositions apart from that areas where Dirac point cold be presence (for x>0.17 for Hg_{1x}Cd_{x}Te alloys as well as for x>0.11 for the Hg_{1x}Zn_{x}Te alloys) an ambivalence behaviour for the temperature dependences of the HgTemodes is observed – could be or strong positive temperature shift or complete independency on the temperature takes place.
3. Discussion
3.1. Basic theory
In view of the most general assumptions, it is possible to start from the following equation for the temperature shift of the TO_{i}phonon mode frequencies ν_{T Oi} [18]:
The first term in Eq. (1) corresponds to the crystal expansion and to an anharmonic contribution to the harmonic crystal potential. This anharmonic term has been analyzed in detail by different authors in the last decades. The theory developed by Maradudin and Fein (Maradudin & Fein, 1962) as well as by [14] is based on the classical anharmonic oscillator. The potential energy is:
where, the cubic term gx^{3} gives a thermal expansion but no change in the frequency at the first order. The quartic term fx^{4} and the cubic term to the second order (gx^{2})^{2} may induce a change in the frequency of the modes. The role of these terms was estimated [10] by assuming for Si atoms a covalent interatomic bond within the Morse potential:
where the constants D, a and r0 are determined, respectively, by bonding energy, stiffness of bond and interatomic spacing. Using the Morse potential to determine the coeﬃcients of the Taylor series expansion of the potential, [14] find that the quartic term is positive, i.e., it increases with the frequency, but accounts for only 3/5 respect to the cubic term to the second order that is negative. The result is then a net decrease in the frequency.
In the quantummechanical approach each power of x correspond to a creation or an annihilation operator for a phonon, and the frequency shift of the optical mode is calculated as the selfenergy of the mode [24]. Using this technique [10] performed detailed numerical calculations for the diamond structure using eigenvectors and eigenfrequencies of the harmonic model deduced by fitting the parameters of the dispersion curves obtained by inelastic neutron scattering data. The appropriate anharmonic interaction was determined by fitting experimental thermal expansion data. Later, Ipatova I.P. et al. (Ipatova et al., 1967) working on ionic crystals and Schall M. et. al. (Schall et al., 2001) in the CdTe and ZnTe semiconductors applied this theory to explain the temperature dependence of the dielectric function in the far IR frequency range.
The second term in equation (1) is due to the ep interaction and it is interesting to underline that an analogue expression takes place for the temperature dependence of the energy gap in semiconductors (Yu & Cardona, 1996) where two contributions also occur: the anharmonic one and that induced by the ep interaction.
3.1.1. Anharmonic contribution
An expression was derived by Ipatova I.P. et al. [16] for the temperature dependence of the phonon mode frequency ν_{T O} or the damping of an oscillator γ_{T O} in the quartic anharmonic force constant approximation:
where Y(T) is one of the measured quantities ν_{TO} or γ_{TO}, Θ is the characteristic temperature of the phonon subsystem Θ=hν_{T O} /k_{B} (k_{B} is the Boltzmann constant), A, B, C are the parameters obtained by fitting and for the CdTe phonon frequency (Ref. 12): A=4.361 THz (or 150 cm^{−1}), B=0.0298 THz (or1.00 cm^{−1}), C=0.0348 THz (or1.16 cm^{−1}). The same parameters for the ZnTe are: 5.409 THz (or 190 cm^{−1}),0.0457 THz (or1.52 cm^{−1}) and0.0341 THz (or1.37 cm^{−1}), respectively. In this case, it is clear that A is the frequency ν_{TO}(0) of the TOphonon mode at T=0 and the first term in the equation (1) can be rewritten as:
Because A and B are usually negative parameters, equation (5) always points out a frequency decrease, i.e., a softening of the phonon frequency on increasing the temperature. This behaviour has been observed in many ionic crystals and widegap semiconductors.
3.1.2. The ep interaction contribution
As mentioned in the Introduction, the returnable ep interaction could be responsible for the abnormal temperature dependence of the phonon frequency in both HgCdTe as well as HgZnTe. This kind of ep interaction induces a discontinuity in the temperature dependence of the phonon frequencies in the resonant case, i.e., Dirac points. Actually, it is possible to assume that far from a Dirac point the returnable ep interaction may overcome the anharmonic contribution and reverse the sign of the phonon frequency temperature dependence associated with the lattice dilatation.
As shown in [19], the preferred mechanism explaining the influence of the electronic structure of the crystal on its phonon spectrum is a deformation potential that mediates the interaction of electrons with the transverse optical phonons (TOphonons). The TOphonons are clearly recognized in optical reflectivity experiments, therefore the deformation potential is responsible for the interaction of electrons with TOphonons. We are interested only in the terms of the deformation potential matrix that correspond to the energy region between the valence and the conduction band. Therefore, the selfenergy of the TOphonons with a small wavevector q is given by the formula [19, 35]:
where E_{g} is the energy gap and
V_{cν} (k, q) does not depend on the wave vector of the longwave optical phonons (q 0); thus, E_{c} (k+q) − E_{ν} (k)=E_{g}.
We can identify two kinds of singularities in Eq. (6): the first is obtained when Eg is equal to, and the second occurs when Eg equals zero. In the second case, if the temperature increases, the E_{g} (T) dependence approaches zero from the negative side of the energy gap (the inversion bandstructure). On the other hand, decreasing the temperature, the E_{g} (T) dependence approaches zero from the positive side of the energy gap (normal band structure), hence, a discontinuity in ω_{T O} (T) may occur also at E_{g} (T)=0.
In order to describe the ep contribution to the phonon frequency temperature dependence we may use an equation derived from Eq. (6) to determine the frequency change:
where Ξ(k, q) is the optical deformation potential, E_{F} is the Fermi energy measured from the band edge, W is the sum of the conduction and the valence bands width,
Looking at ∆ν^{II} (T) we may recognize the sign of this contribution through the corresponding sign of the deformation potential Ξ(k, q) : the sign is ““ when the energy gap (E_{g} ≡ Γ_{6} − Γ_{8}) is positive (usually E_{g} > 0, in a normal semiconductor the deformation potential is negative) or the sign is “+” when E_{g} < 0 (that takes place before a Dirac point). Therefore, before the resonance case (E_{g} ≡ Γ_{6} − Γ_{8}=0) when E_{g} < 0, Ξ(k, q) > 0 and the contribution of the returnable ep interaction to the temperature change of the phonon frequency has a reversed sign with respect to the anharmonic contribution which is always negative. At some temperature a full reverse sign of the frequency could occur, i.e., when the phonon contribution overcomes the anharmonic one, on increasing the temperature also the frequency starts increasing. It enables us to explain the abnormal temperature dependence of the optical phonon frequency.
A different scenario occurs after the resonance: the sign isfor the phonon contribution to the temperature change of the phonon frequency when Ξ(k, q) < 0. The phonon frequency suddenly decreases and a discontinuity takes place at the Dirac point. However, on increasing the temperature the E_{g} ≡ Γ_{6} − Γ_{8} also increases, and the negative phonon contribution to the temperature change of the phonon frequency is reduced that implies a decrease of the negative change of this frequency. Therefore, the magnitude of the phonon frequency increases with the temperature after the Dirac point only if the phonon contribution overcomes the anharmonic one, a condition occurring not far from the resonance. As a consequence, the abnormal temperature dependence of the optical phonon frequency may occur also after the resonance.
3.1.3. Full temperature shift of the TOphonon frequency
According to the above, the full temperature shift of the TOphonon frequency in the semiconductor crystals is ∆ν(T)=∆ν^{Ι} (T)+∆ν^{II}(T) (T) and the temperature dependence of the TOphonon mode ν_{T O} (T) can be written as:
From the comparisons of Eqns (5), (9) and (10) the anharmonic contribution exhibits for all crystals a monotonic function vs. temperature and, because the B and C constants are usually negative, the phonon frequency decreases (Ipatova et al., 1967; Schall et al., 2001). Actually, the ep contribution depends dramatically on the temperature because E_{g}(T) crosses through a point where E_{g}=0. In this point a singularity occurs and, because the ep contribution is huge, a discontinuity in the temperature dependence of the phonon mode frequency is observed for Hg_{0.89} Cd_{0.11} Te at 245 K (Sheregii et al., 2009). When E_{g} < 0, the ∆ν^{II} (T) is positive and leads to the hardening of the phonon mode with increase of temperature. However, this contribution quickly reduces decreasing the temperature. Indeed, when temperature decrease the value of E_{g}  increases and ∆ν^{II}(T) becomes smaller than ∆ν^{I}(T) what means a softening of the phonon mode at low temperatures. In the semiconductor case at (E_{g} > 0), the ∆ν^{II}(T) is negative and ∆ν^{II}(T)  > ∆ν^{I}(T)  is fulfilled, because at E_{g} ~ 0 the ep contribution is large. However, the increase of E_{g}  when temperature increase, leads to a decrease of the total negative change of the phonon frequency that implies a positive temperature shift of the magnitude of the phonon mode frequency already observed in the Hg_{1−x} Cd_{x}Te as well as in another mercury contained alloys. So, it seems to be possible to explain this positive temperature shift of the phonon frequency by the ep contribution Nevertheless, concerning the role of the returnable ep coupling, it is necessary to carry out a reliable experimental test of the above theoretical assumptions on the temperature dependence of the phonon mode frequency.
In what follows, we will present the analysis of the experimental data which are described above. Data allow to answer if the returnable ep coupling contribution is enough to explain the abnormal temperature dependence of the HgTelike mode frequency in both MCT and MZT alloys.
3.2. Analyses and Interpretation of experimental results
3.2.1. Alloy Hg_{0.85}Cd_{0.15}Te
The analysis of the observed temperature dependence of the TOphonon modes for these semimetallic and semiconductor alloys is based on Eq. (10).
In order to calculate the
3.2.2. Alloy Hg_{0.80}Cd_{0.20}Te
In the case of the Hg_{0.80} Cd_{0.20} Tealloy the singularity in the second term of Eq. (10) is very far (formally should exist at temperature closed to absolute zero) and the effect of the returnable ep interaction is negligible. That is shown in Fig. 10b where we show experimental data and theoretical curves of the HgTelike mode. Here the solid curve is calculated with the
The CdTelike mode frequency has an opposite behavior vs. temperature with respect to the HgTelike one: the frequency strongly decreases vs. temperature. It is a classical behavior where the anharmonic contribution dominates in the phonon frequency. The parameters B and C are close to that of the sample Hg_{0.85} Cd_{0.15}Te and also of the binary CdTe.
3.2.3. Alloy Hg_{0.90}Zn_{0.10}Te
To calculate according to Eqns. (9, 10) the
From the above
Really, at this temperature 85 K a discontinuity of the temperature dependences of the HgTelike and ZnTelike modes is observed (see Fig. 5a and 5b). These discontinuities are smaller than in the Hg_{0.85} Cd_{0.15} Te sample. In the case of the HgTelike mode for the sample Hg_{0.9} Zn_{0.1} Te (Fig. 5a),
Eq. (10) describes quite well the experimental behavior with the discontinuity at 85 K but with a value
3.2.4. Alloy Hg_{0.237}Zn_{0.763}Te
The sample Hg_{0.763} Zn_{0.237} Te is characterized by the HgTelike phonon frequency practically independent of the temperature as showed in Fig. 6b, and constants B and C close to zero. The
The temperature dependence of the ZnTelike mode frequency in the case of the Hg_{0.763} Zn_{0.237} Te alloy points out a maximal negative shift among all the alloys we investigated:
3.2.5. The relativistic contribution to vibrational effects
From the above data the positive temperature shift is characteristic in different alloys only of the HgTelike modes. It means that contribution of Hgatoms to the chemical bonds mainly affects the abnormal temperature dependence of the HgTelike phonon frequencies. It is then useful to consider the peculiarities of the chemical bonds in the case of the IIVI compounds with Hg.
Using a simple chemical picture of these compounds, Hg atoms contribute to bonds with two s electrons while Te atoms with two s and four p electrons. In comparison with Ca, Sr, and Ba chalcogenides, the ionicity of Hg chalcogenides is reduced. The Hgdelectrons are partially delocalized, and, therefore, the effective nuclear charge, experienced by the valence electrons, increases. This generates a more tightly bound of Hg valence s electrons and, hence, a less ionic and more covalent bond. In this respect, Hgatoms in the IIVI compounds are similar to the isoelectronic Cd and Zn in the same semiconductors. However, the dshell delocalization is stronger in Hg than in Cd or Zn and, in fact, enough strong to pull the s level below the chalcogen p level10. As a consequence, an inverted band structure is obtained. The role of delectrons in IIVI compounds is discussed in more detail by Wei and Zunger, [42], while the contribution of the spinorbit interaction to chemical bonds and electronic structure is considered in [12]. This contribution increases with the number of atoms, and it is larger for Hg atoms with respect to Cd and Zn ones. In HgTe, the Γ_{8} band is higher in energy than the Γ_{6} one, whereas the situation is reversed in CdTe and ZnTe. Actually, this is because the energy difference between the Γ_{8} and Γ_{6} levels is determined by three factors: i) the chalcogen pspinorbit splitting, ii) the Hgdspinorbit splitting and iii) the coupling strength among these states – socalled pdcoupling. For the p states, the Γ_{8} symmetry is higher in energy than the Γ_{7}, whereas for dstates the situation is reversed. Thus, if the pspinorbit coupling with the Hgdspinorbit split states becomes reasonably small as in CdTe (ZnTe), the order of the Hgdspinorbit split states drives the sequence order of Γ_{8} and Γ_{6} levels. Alternatively, if due to large pdcoupling the d character dominates in these bands, the Γ_{8} level may also end up higher than the Γ_{6} level.
However, how the difference between CdTe (ZnTe) and HgTe electronic structures translates into a temperature dependence of the phonon frequency? As it was underlined in the Subsection 3.1, a similar equation to Eq.(1) takes place for the temperature dependence of the energy gap in semiconductors where two contributions, the anharmonic one and that associated to ep interaction, interplay with each other. It is interesting to note that the energy gap of HgCdTe and HgZnTe alloys have the same positive shift (Dornhaus & Nimtz, 1985) as the HgTelike modes frequencies, actually opposite to the temperature shift observed for the energy gap of both CdTe and ZnTe [41, 23]. This difference in the E_{g}(T) dependences for binary CdTe(ZnTe) and ternary HgCd(Zn)Te is due to the Hgdspinorbit split states and, probably, is translated into the ν_{TO}(T) also due to a large pdcoupling of the chemical bonds. Although, the temperature dependence of the expansion coeﬃcients in CdTe(ZnTe) and HgTe are similar (Bagot, 1993), the role of the lattice expansion on the ν_{TO}(T) dependences is different, similarly to the E_{g}(T). It is possible to claim that the role of the quartic term in the potential of the Eq.(2) (see Subsection 3.1) is different for a HgTe bond compared to CdTe and ZnTe ones because of the Hgdspinorbit split states: overcomes others terms and positive temperature shift of the phonon frequency takes place.
4. Conclusion
It was performed an extensive experimental investigation of the temperature dependence of the phonon mode frequencies for Hgbased semiconductor alloys of IIVI compounds using the synchrotron radiation as a source in the farinfrared region. In the case of the Hg_{0.9} Zn_{0.1}Te alloy we found a discontinuity of the temperature dependence of HgTelike T0mode and ZnTelike T_{1}mode similarly to the Hg_{0.85} Cd_{0.15} Te alloy firstly found five years ago by [35]. A theoretical expression (Eqn. (10) in subsection 3.1) for the temperature shift of the phonon mode frequency has been derived including an anharmonic contribution as well as a term of a returnable electronphonon interaction. It was shown that this expression including both abovementioned contributions satisfactorily describes the temperature shift of Hg_{0.85}Cd_{0.15}Te and Hg_{0.90} Zn_{0.10} Te alloys containing a Dirac point (E_{g} ≡ Γ_{6} − Γ_{8}=0) if one of the two constants B and C describing the anharmonic shift of the HgTelike mode, is positive. Moreover, in the case of the semiconductor alloys Hg_{0.80}Cd_{0.20}Te and Hg_{0.763} Zn_{0.237} Te the role of the returnable
The difference between the temperature behaviour of HgTelike modes and CdTeor ZnTelike ones can be explained by the Hgdspinorbit split contribution to the chemical bond. This contribution is responsible of the positive temperature shift of the energy gap of ternary HgCdTe and HgZnTe alloys with a narrow gap because the relativistic contribution to chemical bonds is also at the origin of the abnormal temperature shift of electron states in Hgbased semiconductors – inverse band structure. Similar effect is reasonably expected that the Hgdspinorbit split contribution leads to an abnormal temperature shift of the HgTelike phonon mode frequency.
Acknowledgments
Author is greatly indebted to staff of the Laboratori Nazionale di Frascatti for possibility to perform several Project in framework of the TARIcontract in the years 2002 – 2006. This work was partly supported by the EU Foundation by the TARIcontract HPRICT199900088.
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