Electronic, vibrational and thermodynamic properties of -HgS(metacinnabar)

M. Cardona, R.K. Kremer, R. Lauck and G. Siegle

Mercury sulfide HgS is usually found in na-ture as red cinnabar (-HgS) which crystallizesin the Se-like trigonal structure which containsangulate Hg–S–Hg chains. Cinnabar constitutesthe standard ore for the smelting of elementarymercury. Black HgS, meta-cinnabar, (-HgS)which crystallizes in the cubic zincblende struc-ture, can be obtained by precipitation in the lab-oratory as a fine polycrystalline powder whichcan be easily converted (e.g., by chemical va-por transport) into red cinnabar. Addition ofsmall amounts (≈ 1%) of Fe helps to crystal-lize -HgS in the zincblende structure both innature, and in the laboratory.

While the electronic and thermal properties of-HgS appear to be rather well investigatedthose of -HgS are less well-known. In thiswork we describe the vibrational and thermalproperties of -HgS which we carried out bythree different approaches [1]: an experimen-tal one and two ab initio electronic structurecalculations using different implementations ofdensity functional theory. In order to obtainthe electronic structure, we have employed theABINIT and VASP codes with inclusion ofspin-orbit (s-o) interaction. Exchange and cor-relation effects have been taken care of bythe two well-known approximations, LDA (lo-cal density approximation) and the PBE-GGA(Perdew, Burke, Ernzerhof generalized gradientapproximation).

Additionally, we have obtained the phonon dis-persion curves in a quasiharmonic approxima-tion and calculated the heat capacity CV whichwe have compared with our new experimen-tal results. The experimental data obtained fornatural -HgS crystals (originating from MountDiablo Mine, Clayton, CA; (1.1 mass-% Fe ac-cording to chemical microanalysis) and on ahigh purity (99.999%) fine powder sample. Fi-nally, we have also performed calculations ofthe enthalpy of HgS versus pressure for the

zincblende, rocksalt and cinnabar phases. From

the dependence on pressure we are able to de-

rive critical pressures at which phase transitions

will occur and compared these with experimen-

tal results.

Figure 1: (a) Electronic band structure of -HgScalculated with the ABINIT code in the LDA ap-proximation and with s-o interaction using the lat-tice parameter a0 = 5.80 A obtained by total energyminimization. (b) Electronic band structure with theenergy scale enlarged so as to display the details ofthe top valence bands, including s-o splitting, andthe lowest conduction band around . The 8–7separation corresponds to the negative s-o splitting0. 6–8 is the E0 gap. We should mention thatFigs. 1(a) and (b) are affected by the so called gapproblem which incorrectly lowers the 6 state withrespect to the (7–8) states. Recent unpublishedcalculations by Axel Svane suggest that this low-ering may be ≈ 1 eV. Thus correcting this effect,as done when using the GW approximation mayrestore the conventional conduction (6)-valence(8–7) band ordering.

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Figure 1 displays the band structure of -HgSas calculated with the ABINIT code using theLDA approximation with the energy optimizedlattice parameter a0 = 5.80 A. The wide energyscale chosen in Fig. 1(a) allows us to observethe position of the lowest, sulfur 3s-like 1 va-lence band, which is not affected by s-o inter-action, at about –12 eV. Above this band, the5d bands of Hg appear. Without s-o interactionthey split into an upper quadruplet (≈ –6 eV,12) and a lower sextuplet (15). s-o interac-tion does not split the 12 states but it splits the15 sextuplet into a doublet 7 and a quadru-plet 8, the former being above the latter. Thesplitting shown in Fig. 1(a) is 2.1 eV , the sameas the corresponding atomic splitting of Hg [2].At –0.58 eV we find an s-like 1 state which iscomposed mainly of 6s states of Hg. This statecorresponds to the lowest conduction band formost tetrahedral semiconductors above the 15valence bands.

In order to resolve the details of the band struc-ture around the point we replot in Fig. 1(b) theelectronic structure around with an expandedenergy scale. This figure shows the inverted s-ointeraction at – the 7 doublet is above the 8quadruplet. This fact arises from the contribu-tion of the negative s-o splitting of the 5d 15states of Hg which overcompensates the posi-tive splitting of the S 3p contribution. Countingbands we see that the 8 bands should be occu-pied whereas the s-o split 7 should be empty.Hence, according to Fig, 1(b), undoped (intrin-sic) -HgS should have an indirect energy gapof≈ 0.15 eV and a direct one of 0.18 eV. This isin reasonable agreement with the gap observedby Zallen and Slade (0.25 eV), although theseauthors did not realize that the s-o splitting of-HgS is negative [3].

A few Inelastic Neutron Scattering (INS) de-terminations of the phonon dispersion relationsof -HgS [4], have been published. They havebeen compared with semiempirical calculationswhich use fitted force constants. Here we com-pare the measured points with ab initio calcu-lations and examine the effect of s-o interac-tion on these calculations. Figure 2(a) displaysthe phonon dispersion relations of -HgS cal-culated with s-o interaction (solid line) and also

without (dashed line). The effect of the s-o in-teraction (inset for the TA phonons) is rathersmall, however it is possible to see that for theTA phonons, those mainly responsible for theCv to be discussed later, the frequencies with-out s-o are about 5% lower than with s-o.

Figure 2: (a) Phonon dispersion relations of - HgScalculated with the ABINIT-LDA code with (solidline) and without (dashed line) s-o coupling. Thecircles were obtained with INS [4]. The LO andTO phonons at k≈ 0 (red) squares were obtained byRaman scattering. The frequency scales in the insetshave been expanded to show the effect of the s-oaround X. (b) Phonon density of states of -HgS,as obtained with the ABINIT code. We have plottedthe total DOS and also their projection on the Hgand the anion atoms. Note that in the case of -HgSthe acoustic phonons correspond to nearly pure vi-brations of Hg, the optic ones to nearly pure S vibra-tions.

An interesting feature is the strong upwardsbending of the optical bands away from k= 0,the opposite of what happens for most tetrahe-dral semiconductors. This effect is due to thelarge mass difference of Hg and S. At the edgeof the zone the ‘effective’ mass which deter-mines the frequency is basically that of sulfur,whereas at k= 0 it is the somewhat smaller re-duced mass. In order to obtain the calculatedup-bending one must also invoke a strong forceconstant connecting the anions.

The upward bending is reflected in the corre-sponding density of phonon states, shown inFig. 2(b). A direct consequence is that the TORaman modes do not overlap the DOS con-tinuum while the LO mode does. The width ofthe former should therefore not be affected byelastic scattering due to defects, such as isotopicfluctuations, whereas the LO-modes, overlap-ping the background, should.

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The up-bending of the dispersion relations isalso important when considering confinementin quantum dots. These effects have been dis-cussed for the case of CdS, a material for whichthe TO bulk bands bend up but the LO bandsbend down [5]. No such experiments are avail-able, to the best of our knowledge, for -HgSbut quantum confinement in dots of this mate-rial should produce an increase in frequency asopposed to most other related materials.

Figure 3: (a) Cp/T3 measured on a mineral and apowder sample of -HgS. Data by Khattak et al. arealso given for comparison [6]. The solid line repre-sents the mineral data minus a linear contribution asindicated in the inset. (b) Same mineral data as be-fore compared with the results of four calculationsas indicated.

Calculations of Cv were performed by inte-grating the calculated phonon DOS with thestandard expression. The calculations were per-formed both, with the ABINIT-LDA and theVASP-GGA codes, with and without s-o inter-action. The effect of s-o interaction was basi-

cally the same for the two codes – the s-o inter-action lowered the maximum ofCv/T3 by about10%. We shall present here the results obtainedwith only one of the codes (except for -HgS)but both, with and without the s-o interaction.We show in Fig. 3(a) the experimental valuesof Cv/T3∗). for two sample of -HgS, a natu-ral crystal and a high-purity powder. The natu-ral crystal was found by chemical microanalysisto have an Fe content of 1.1mass-%. The up-turn at low temperatures observed for the natu-ral crystal can be ascribed to a linear term in thespecific heat (Sommerfeld term) due to metal-lic behavior. To correct for this term we sub-tracted a linear contribution = 20mJ/molK2)from the experimental data. Figure 3(b) displayscalculated VASP and ABINIT values of Cv/T3

for -HgS obtained with and without s-o inter-action. The agreement between theoretical andexperimental data is reasonably good. Our newmeasurements seriously question the heat ca-pacity data by Khattak et al. [6] shown for com-parison in Fig. 3(a).

Figure 4: Enthalpy of the three phases of HgS cal-culated with the ABINIT-LDA code with s-o inter-action. Cinnabar and zincblende phases are basicallydegenerate (within any reasonable error estimates)at p= 0 although formally their enthalpies cross at≈ 0.7GPa. The cinnabar → rock salt transition ispredicted to occur at ≈ 22GPa. Experimental data(not shown) place it at 20.5GPa [8]. For the sakeof clarity the cinnabar enthalpy has been subtractedfrom all curves.

∗Cp represents the experimentally measured value of the heat capacity at constant pressure. The ab initiocalculations yield Cv, i.e., the heat capacity at constant volume. In the temperature region considered hereCv≈Cp.

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Mercury chalcogenides undergo phase transi-tions at relatively low pressures. While there isconsiderable experimental data on these tran-sitions [7], no ab initio calculations seem tohave been reported. We have performed en-thalpy minimizations vs. pressure for HgS con-sidering the three most conspicuous structuresof these materials: cinnabar, zincblende androck salt. We show in Fig. 4 the pressure de-pendence of the enthalpy of the three phasesmentioned, as calculated for HgS. The cinnabarphase (-HgS) is found, within estimated er-ror, to have the same enthalpy as the zincblendephase (-HgS), a fact which may explain theease to obtain both phases in the laboratory, al-though the growth of the zincblende phase re-quires the addition of a small amount of Fe.

In summary, our ab initio calculations of theelectronic band structure and the phonon dis-persion relations of the zincblende-type -HgSpredict a negative spin-orbit splitting which re-stores semiconducting properties to the materialin spite of the inverted gap. We have obtainedthe spin-orbit induced linear terms in k whichappear at the 8 valence bands (not shown here,for details see [1]) and investigated the pres-sure dependence of the crystal structure and thephonons. The temperature dependence of thespecific heat capacity has been calculated and

first conclusive low-temperature experimentaldata on -HgS have been presented.

In Collaboration with:A. Munoz (MALTA Consolider Team, Departamento deFısica Fundamental II, and Universidad de La Laguna,Tenerife, Spain); A.H. Romero (CINVESTAV, UnidadQueretaro, Queretaro, Mexico)

[1] Cardona, M., R.K. Kremer, R. Lauck, G. Siegle,A. Munoz and A.H. Romero. Physical Review B 80,195204 (2009).

[2] Herman, F. and S. Skillman. Atomic Energy Calcula-tions, Prentice Hall, Englewood Cliffs, NJ (1963).

[3] Zallen, R. and M. Slade. Solid StateCommunications 8, 1291–1294 (1970).

[4] Szuszkiewicz, W., B. Witkovska, M. Jouanne. ActaPhysica Polonica A 87, 415–418 (1995);Szuszkiewicz, W., K. Dybko, B. Hennion, M. Jouanne,C. Julien, E. Dynowska, J. Gbrecka andB. Witkowska. Journal of Crystal Growth 184–185,1204–1208 (1998).

[5] Chamberlain, M.P., C. Trallero-Giner andM. Cardona. Physical Review B 51, 1680–1693(1995).

[6] Khattak, G.D., H. Akbarzadeh and P.H. Keesom.Physical Review B 23, 2911–2915 (1981).

[7] Mujica, A., A. Rubio, A. Munoz and R.J. Needs.Reviews of Modern Physics 75, 863–912 (2003).

[8] Nelmes, R.J. and M.I. McMahon. Semiconductorsand Semimetals 54, 145 (1998); (Eds. T. Suski andW. Paul). Academic, New York.

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