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Birefringence and band structure of CdP2 crystals
Physica B 422 (2013) 12–19
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Birefringence and band structure of CdP2 crystals S.I. Beril a, I.G. Stamov a, N.N. Syrbu b,n, V.V. Zalamai c a b c
Tiraspol State Corporative University, Yablocikin Street 5, 2069 Tiraspol, Republic of Moldova Technical University of Moldova, 168 Stefan cel Mare Avenue, 2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Street, 2028 Chisinau, Republic of Moldova
art ic l e i nf o
a b s t r a c t
Article history: Received 12 February 2013 Received in revised form 4 April 2013 Accepted 6 April 2013 Available online 20 April 2013
The spatial dispersion in CdP2 crystals was investigated. The dispersion is positive (nk||с 4nk||у) at λ4 λ0 and negative (nk||с o nk||у) at λ oλ0. CdP2 crystals are isotropic for wavelength λо ¼ 896 nm. Indirect transitions in excitonic region Еgx are nonpolarized due to one pair of bands. Minimal direct energy intervals correspond to transitions Г1-Г1 for Е||с and Г2-Г1 for Е⊥с. The temperature coefficient of energy gap sifting in the case of temperature changing between 2 and 4.2 K equals to 10.6 meV/K and 3.2 mev/K for Г1-Г1 and Г2-Г1 band gap correspondingly. Reflectivity spectra were measured for energy interval 1.5–10 eV and optical functions (n, k, ε1, ε2, d2ε1/dE2 and d2ε2/dE2) were calculated by using Kramers–Kronig analyses. All features were interpreted as optical transitions on the basis of both theoretical calculations of band structure. & 2013 Elsevier B.V. All rights reserved.
Keywords: Semiconductor compound Optical absorption and reflection spectra Kramers–Kronig relations Optical constants Excitons Band structure
1. Introduction The cadmium diphosphide is a wide-gap AIIBV semiconductor and possesses the anisotropy of optical properties with natural gyrotropy [1–3]. The optical spectra of photoluminescence and photoconductivity, transmission and reflection for the region of electronic transitions and vibrational modes are polarized [3–12]. Quantum electronic and nonlinear optic devices were developed on the basis of CdP2 crystals whose operating principle was based on gyrotropy and nonlinear polarizability of the crystal. Values of nonlinear polarizability and gyrotropy of cadmium diphosphide are higher than other crystals and they provide effective stabilization of radiation field in the space (lateral section) and in the time simultaneously [13]. Stabilizers of emission field and extenders of lasers impulse duration on the basis of CdP2 were developed. The low thermal-conductivity (10 W/mK) of CdP2 crystals was used for producing laser beam deflectors with thermal induced gradient of refractive index. Advantage of deflectors is a possibility of their usage in visible and infrared regions [14,15]. The linear dependence of polarization plane of plane-polarized light on magnetic field induction was found in cadmium diphosphide and a possibility of producing on the basis of CdP2 of magnetooptical modulators and magneto-optical sensible elements of measuring of magnetic field induction was shown [16,17].
n
Corresponding author. Tel.:+373 22 237508. E-mail address: [email protected] (N.N. Syrbu).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.04.029
The cadmium diphosphide has a high photosensitivity and intense luminescence [3–5]. The technology of p–n junction producing surface-barrier diodes was developed on the basis of CdP2 crystals [2,3,5,7,8]. CdP2 based photoresistors have small relaxation time and are suitable for registration of pulse radiation with duration of 1 ns. Photoelectron emitters, photodiodes, electrical switchers and stabilitrons [18–20] were developed on the basis of CdP2 monocrystals. CdP2 crystals change the optical activity with temperature change at the same time the linear temperature dependence of rotatory power of polarization plane which remains invariable. Temperature sensors were developed due to this effect [20]. Last year the level of technology allows to grow more perfect and high quality crystals. In this work properties of birefraction, luminescence, and absorption in the region of band gap minimum were investigated for perfect crystals and new experimental results were received. Reflection spectra were researched for wide energy region (1–10 eV), calculated using Kramers–Kronig relations and discussed based on two variants of band structure theoretical calculations. 2. Experiment The process of diphosphide monocrystal growing from precursors Cd and P had two stages. The purifications initial substances and synthesis of CdP2 chemical solution were carried out during the first phase. The synthesis was executed in quartz ampoules which were deposited in high pressure (40 atm)
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chamber. Up to 300–500 g of material was synthesized simultaneously. During second stage the synthesized substance was treated in vacuum for receiving monocrystals. The maximal size of ingots of CdS2 grown from gas phase achieved 15 15 30 mm. Some single crystals had clear crystallographic shape and did not need subsequent treatment and orientation according to crystallographic axes. Monocrystals were cleaved out perpendicular to caxis. Measurements were executed for monocrystals grown along c-axis as a prism or plate with smooth nontreated surfaces. Measurements at E||c and Е⊥с, k||у polarizations were carried out on nontreated smooth surface (1 0 1). In the case of Е⊥с and k||с polarization samples cleaved out perpendicular to c-axis from the same ingot were used. Absorption spectra at 2 K and 4.2 K were measured in glass cryostat with liquid helium by means of spectrometer DFS-24 with resolution 5 Å/mm. The spectral width of spectrometer slit during the measurements of absorption spectra was less than 0.1 Å. Optical spectra of absorption and luminescence at 9 K were measured in cryostat LTS-32C330 Workhorse on spectrometer DFS-32 with the same value of spectral slit of spectrometer. Reflection spectra were measured by spectrometer Specord M40 at 77 K in energy range of 1–6 eV.
3. Results and discussion 3.1. Birefringence of CdP2 crystals The spectral dependence of transmission coefficient (T) of CdP2 crystals that deposit between two crossed polarizers is shown in Fig. 1. The position of polarizers and crystal axes orientation during measurements can be seen in inset. It was found that the maximum at wavelength 654 nm in transmission spectra of crystals with thickness d ¼ 1117 mm deposed between crossed polarizers (Fig. 1, curve a). Additional peaks appear at short and long wavelength parts of spectra by changing of crystal thickness to 739 and 470 mm, but at the same time the central line at 654 nm
Fig. 1. The spectral dependence of transmission (T) of CdP2 crystals with thicknesses 1.117 mm (a), 0.73 mm (b) and 0.47 mm (c) in crossed polarizers. The inset shows the orientation of crystal axes.
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remain in spectra (see Fig. 1, curves b and c). The wavelength 654 nm is isotropic wavelength (λ0) for CdP2 crystals. The crystal does not recognize a polarization of light waves in this wavelength. The Fig. 1 shows the spectral dependence of transmission T in diapason from 0.6 to 2.5 mm for CdP2 crystals with thickness 1.117 mm (a), 0.73 mm (b) and 0.47 mm (c) deposed between crossed polarizers. The clear interference with lines intensifying in short wavelength part of spectra is observed. The difference between refractive indexes Δn ¼no−ne was defined by these dates. The curve Δn intersects the wavelength axis at isotropic wavelength λо (654 nm). This difference Δn¼ no−ne achieves minimal value (−0.3) at 2.0 mm wavelength. Reflectivity spectra for short wavelength part from isotropic wavelength were measured and refraction indexes for different polarizations of light wavelength E and direction of wavevector k were calculated by Kramers–Kronig (KK) relations (see Fig. 2b). Measurements of reflectivity spectra were carried out for two surfaces of crystal (1 1 0) and (1 0 1) almost at angles close to normal (Fig. 1). For determining parameters in Е||с, k||y, Е⊥с, k||y polarization light beams were directed on sample surface ac(xy). The electrical vector of light wave allocated in the crystallographic plane ac(xy) and the wavevector k was perpendicular to axis c (k⊥c). For polarizations Е||y, k||c, Е||x, k||c light rays were directed on surface ab(xy) parallel to the optical axis c. At sample living the plane of vibrations of electrical vector turns on angle (φ¼ ρd, where ρ—specific turn of vibrations plane) due to natural optical activity at low intensity of light. Spectral dependences of refractive indexes for polarizations Е|| с, k||y, Е⊥с, k||y и Е||y, k||c, Е||x, k||c calculated from reflectivity spectra by Kramers-Kronig relations were shown in Fig. 2. Refractive indexes for polarizations Е||y, k||c, Е||x, k||c change almost
Fig. 2. (A) Fragments of interference in transmission spectrum (T) of crystal CdP2 in crossed polarizers and spectral dependence Δn ¼no–ne, and (B) spectral dependences of refractive indexes no, ne и Δn¼ no–ne determined by Kramers–Kronig analyses.
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parallely. Also in polarization Е||с, k||y, Е⊥с, k||y the character of changing of refractive indexes is parallel, but they cross at energy 1.879 eV (896 nm). This cross-point is a isotropic point with wavelength λ0 for CdP2 crystals. The transmission maximum at this wavelength in transmission spectra of crystals at crossed polarizers is observed (Fig. 1, curve T). The value of refractive index nk||с is more than refractive index in polarization nk||у in longwavelength part of λ0, it means that dispersion is positive. The opposite dependence observes in short-wavelength region—the dispersion is negative. Such crystal represents a phase plate in which two light waves propagate with rates Vx ¼c/nk||с and Vy/nk||у. The spectral characteristic of the difference of refractive indexes Δn ¼no−ne determined by KK analyses and interference match and both curves have a zero value at 654 nm wavelength. At high intensity of radiation the additional turn of vibrational plane on angle γЕ02, whose value is proportional to light intensity, is observed [13]. In work [13] the efficiency of induced gyration by the dependence changing of transmission intensity of system polarizer-sample-analyzer was estimated. Since measured crystals take place linear and nonlinear losses of radiation thus the intensity of light after analyzer was determined experimentally. It was shown [13] that the dependence of specific value of angle φ/ d induced by nonlinear rotation of polarization plane of light in CdP2 crystals has been proportional to the intensity of light J0. The edge absorption in CdP2 crystals at room temperature has a gentle increase of absorption coefficient for both polarizations (Е|| с, Е⊥с). The temperature decrease causes a shift of absorption edge in short-wavelength region. At low temperatures in region of small absorption coefficients in spectra appear features in Е⊥с and Е||c polarizations which correspond to indirect transitions with phonon emission. These features one can see clearly in spectra d2K/ dE2; they have been marked in Fig. 3 as ξ1–ξ4. The position of ξ1– ξ4 maxima almost coincide in both polarizations. The threshold value of nonphonon indirect transition in excitonic band Еgx is equal to ξ1–Еf, where Еf is energy of phonon participated in
indirect transitions. The minimal energy of phonons 5.4 meV (43.5 cm−1), 9.5 meV (77 cm−1), 9.72 meV (78.4 cm−1), 9.97 meV (80.4 cm−1) by Raman analyses of CdP2 crystals was determined. It turned out almost all vibrational modes in high energy part of spectra. Indirect phonon-aided transitions begin with participation of low-energy optical phonon with energy 5.4 meV (43.5 cm−1). In this case the indirect energy gap in excitonic band Еgx equals to ξ1 (2.16 meV)–Еf (5.4 meV) ¼2.1547 eV. The threshold value of nonphonon indirect transition to excitonic band coincides for polarizations Е⊥с and Е||c and equals to 2.1547 eV with accuracy about 0.0005 eV. CdP2 crystals have a structure described by space group D84 (D44 ). Eight formulaic units compose a unit cell i.e. 24 atoms and amount of phonons in usual case is equal to 72. The presence of such amount of vibrational modes allows observing multiphonon emission of free excitons and indirect transitions with multiphonon absorption and emission. Threshold values of indirect excitonic transitions and phonon energy are presented in work [2]. Intensive lines Еgx1 and Egx0 (2.1533 eV and 2.1547 eV) were discovered in luminescence spectra of CdP2 crystals measured at 9 K and excited by 4765 Å lines of Ar laser. Lines a1–a17 were observed in long-wavelength part from these lines (see Fig. 4). The energy of maximum Egx0 (2.1547 eV) matches with the energy of indirect transition in excitonic band. Exactly this value was received from spectra of modulated absorption d2K/dE2. The energy distance between emission lines Еgx1 and Egx0 equals 1.4 meV. This splitting causes by the splitting of excitonic states due to the exchange interaction. Maxima a1–a17 correspond to phonon emission at free indirect exciton annihilation. Table 1 shows an energy position of discovered luminescence lines and participated phonons. Direct transitions are observed in energy region higher than energy of indirect transition (2.1534 eV). Spectral characteristics of edge absorption in polarizations E||c and E⊥c at temperatures 300, 80, 4.2 and 2 K do not coincide (Fig. 5). Edges of absorption intersect at the wavelength λ0 for the diapason of absorption coefficients 200–400 cm−1. The edge of absorption for E⊥c is shifted to low-energy part in comparison with E||c polarization for all high values of absorption coefficients. The sharp increasing of absorption coefficient for this diapason is caused by direct allowed transitions. Calculated values of absorption coefficient coincide with experiment at appropriate parameters: Еgx ¼ 2.2727 eV and А¼3.04 104 eV1/2/сm and Еgx ¼ 2.3078 eV and А¼2.8 104 eV1/2/сm at temperature 2 K and polarization Е⊥с and E||c correspondingly, where A¼
2e2 ð2mnÞ3=2 P CV j2 m20 cℏ2 n
ð1Þ
Received results show that absorption curves are shifted on 32.1 meV in region of direct allowed transitions at 2 K for E||c and
Fig. 3. Absorption spectra K and their energy derivatives d2K/dE2 of CdP2 crystals measured at 9 K.
Fig. 4. Luminescence spectra of CdP2 crystals measured at 9 K and excited by 476.5 nm Ar laser line.
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Table 1 Maxima of emission spectra of CdP2 crystals at 9 K. No. ai
ai
Egx (2.1534)–ai, meV
IR, R, meV (Г2, А2) [4], [2]*
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17
2.1440 2.1419 2.1386 2.1356 2.1274 2.1253 2.1209 2.1190 2.1150 2.1106 2.1073 2.1049 2.1002 2.0961 2.0945 2.0922 2.0889
9.4 11.4 14.7 17.7 25.9 28.0 32.4 34.3 38.4 42.7 46.0 48.4 53.1 57.2 58.8 61.1 64.5
9.5* 11.4* 14.3 17.5* 26.16 11.4+17.7 14.7+17.7 28.0+6.2* 38.1 28+14.7 28+17.7 9.4+39.3 55.5
Г5 [2]*
26.2
38.1
55.4 57.5
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of valence and conduction bands is observed i.e. then k ¼0. The band symmetry according to selection rules for direct electronic transition was established in Г point [2,23]. In case of E⊥c polarization selection rules allow transition of symmetry Г1,2,3,4-Г5, Г5-Г1+Г2+Г3+Г4 and transitions of symmetry Г1,2(s)-Г1(s)+Г2(s) which permit taking into account a spinorbital interaction. The minimal direct energy interval (2.2727 eV) is therefore observed at E⊥c polarization due to transitions Г1-Г1 or Г2- Г2. For E||c polarization transitions between bands with symmetries Г1,2,3,4-Г2,1,4,3, Г5-Г5 and Г1,2(s)-Г1,2(s) are allowed by selection rules. The minimum of conduction band has symmetry Г1 according to all calculated models of band structure. Therefore the upper valence band V1 has a symmetry Г1 in point Г and V2 band has Г2 symmetry (see Fig. 4). The top valence band V1 in point Z has symmetry Z1. Indirect transitions from band Z1(V1) to band Г1(С1) are allowed for both polarizations but they are not observed experimentally.
58.0 38.4+25.9
3.2. Structure of energy bands for CdP2-D48 crystals Reflectivity spectra in the depth of fundamental absorption of CdP2-D48 crystals at temperature 77 K for polarizations E||c and E⊥c for energy interval 1–6 eV and at 300 K for energies 6–10 eV were measured (see Fig. 6A). The absorption coefficient K was calculated for both polarizations from measured reflection spectra by using Kramers–Kronig equations (Fig. 6B). At intrinsic zone (E 1.5–10 eV) in reflection spectra maxima a1–a14 for E||c polarization and e1–e16 for E⊥c polarization are
Fig. 5. Absorption spectra of CdP2 crystals with thickness 7.3 mm in polarized light measured at 4.2 and 2 K temperatures.
E⊥c polarizations, thus direct minimal transitions come from two valence bands. These transitions are allowed in different polarizations by selection rules. The temperature increasing from 2 to 4.2 K causes the long-wavelength shift of absorption edge. The temperature coefficient of energy interval shift ΔE=ΔT stipulated by allowed direct transitions is equal to 10.6 meV/K and 3.2 meV/K for polarization E||c and E⊥c correspondingly at temperature changing from 2 to 4.2 K. Different coefficients of edge absorption temperature shift for E||c and E⊥c polarizations argue whether direct transitions come from two bands V1 and V2. Received experimental results give evidence that band gap Eg1ind corresponds to indirect transitions. According to calculations of band structure of CdP2 crystals the maximum of valence band is localized in point Z of Brillouin zone [21,22]. Therefore indirect transitions occur between extremum Г–Z (Z–Г) in E||c and E⊥c polarizations. The fragment of band structure in band minimum is shown in inset of Fig.5. Indirect transitions are nonpolarized since these Г–Z transitions with phonon participation are allowed by selection rules for both polarizations [2]. Direct transitions take place without phonon and thus they are polarized [2,23]. In the centrum of Brillouin zone without taking into account spin-orbital interaction occur next dispersion laws Е(k,ε) ¼ αkZ2+bk┴2+eεzz+dε┴ ¼Δ1; (k┴2 ¼kх2+kу2, ε┴ ¼εхх+εуу) and with taking into account spin–orbital interaction Е(k,ε) ¼Δ1+ι(kх+kу)7 (m2kz2)1/2 [23]. Dispersion laws are presented for values of wavevector where according to theoretical calculations extremum
Fig. 6. Reflection spectra of CdP2 crystals measured in polarized light (1–6 eV) at 77 K and in nonpolarized light (6–10 eV) at 300 K. The inset shows a model of band structure for band-gap minimum.
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observed. The absorption coefficient has a high value in E||c polarization for all measured diapason and changes in limit of (2.5–5.5) 105 cm−1. Fig. 6 shows that the absorption for intrinsic zone in polarization E||c is less than in polarization E⊥с. Optical functions n, k, (Fig. 7) ε1, ε2, d2ε1/dE2 and d2ε2/dE2 (Figs. 8 and 9) were calculated for all measured diapason in both polarizations by Kramers–Kronig analyses. Maxima of reflectivity spectra and optical factions (n, k, ε1, ε2 and d2ε2/dE2) were clearly recognized at 77 K and also at 300 K (see Fig. 10). Features observed in reflection spectra and optical functions are caused by direct optical transitions in actual points in Brillouin zone. The intensive picks observed at 77 K were detected also at room temperature in reflection spectra and thus in optical function
Fig. 9. The spectral dependence of optical functions ε2, d2ε2/dE2 and ε1, d2ε1/dE2 of CdP2 crystals for E||c measured at 77 K. Fig. 7. The spectral dependence of optical functions n and k calculated by Kraners– Kronig analyses from reflectivity spectra of CdP2 crystals in E||c and E⊥c polarizations measured at 77 K.
Fig. 8. The spectral dependence of optical functions ε1, d2ε1/dE2, and ε2, d2ε2/dE2 of CdP2 in E⊥c polarization measured at 77 K.
spectra. These features are separated in to four groups of maxima Еg–а2 (е2), а3 (е3)–а8 (е8), а9 (е9)–а12 (е12) and high-energy group а13 (е13)–а16. These groups of transitions come from extremum of valence bands to extremum of conduction band at the same value of wavevector k. The Brillouin zone is a rectangular prism with 14 singular points [23]. Group-theoretical calculations of dispersion laws in actual points of Brillouin zone and calculation of selection rules of CdP2 for all points of Brillouin zone with and without taking into account the spin-orbital interaction were carried out [2,23]. Actual points of Brillouin zone (points of zero slope of energy), dispersion laws and selection rules in these points were defined. The zero energy slope realizes by two of three directions kx, ky, kz for Г, Z, М, R, Σ, Λ, N, X, R, А points. For all three directions the zero slope of energy realizes in points Г, М, R without taking into account a spin-orbital interaction. In the case of taking a spin into account the zero slope of energy for all directions fulfils in Г point. Theoretical calculation of band structure show extrema exactly in Г, Р, Z and V points of Brillouin zone [21,22]. The ratios of dispersion analyses show that they have the simplest view in Г, X, P, N, Z points. Transitions in R, X, Y, Т, S and Σ points are not polarized according to selection rules. Selection rules in centrum of Brillouin zone indicate that direct transitions have a different energy interval for each polarization that is polarized. Theoretical calculations of band structure at a wide energy diapason into different points of Brillouin zone for cadmium phosphide with D2h20 symmetry were executed by pseudopotential
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Fig. 11. Electronic transitions into the structure of energy bands of CdP2 crystals [23,24] and Brillouin zone (insert) of crystals with D48 symmetry.
Fig. 10. The spectral dependence of optical function ε2 and its derivation d2ε2/dE2 of CdP2 crystals measured at 300 K for E⊥c (A) and E||c (B) polarizations.
method [24,25]. The crystal lattice of Cd3P2 was considered as an analog of lattice with structure of fluorite during theoretical calculations. Cd atoms are introduced as vacancies periodically deposed in lattice of crystal. The maximum of valence band and the minimum of conduction band are found in Г point of Brillouin zone center. The spin–orbital interaction and another relativistic effect are not taken into account during calculation. The maximum of valence band in Г point is triply degenerated. Taking into account the crystal field potential and spin-orbital interaction lead to the valence band splitting in the case of real crystals. The maximum of valence band Г15 splits in three bands due to degeneration removing. Valence bands have maxima and conduction bands have minima at the same wavevector k in Г, Х and L points. Theoretical calculations of band structure of CdP2 and ZnP2 crystals executed by pseudopotential method in Г, Z, V, U, R, А, Р, М and Λ points of symmetry at wide energy diapason (25 eV) were described in works [21,22]. A huge amount of bands with flat character was received. The lowest conduction band as well as calculations presented in [24,25] is single and its minimum situated in Γ point. Valence bands are also flat. The work [26] shows calculations of band structure of ordered chalcopyrite ZnSnP2 and disordered structure of sphalerite Zn(Sn)P2 in Г, Х, Р, N and Z points of Brillouin zone. The lower conduction band is formed by s-states of Zn atoms and the upper valence band is formed by p-states of phosphor atoms. Received band diagrams of Zn(Sn)P2 [26] and Zn3P2 [24,25] crystals is similar and do not contradict with the results for CdP2, ZnP2 [21,22]. In this work experimental results are discussed in framework of band model [21,22,24,25] subject to dispersion laws into the points of extrema localization. In the structure of energy bands presented in [21,22]
it is a bit difficult to recognize values of wavevector where maxima of valence band and minima of conduction band coincide. For comparison of experimental and theoretical energy gaps the theoretical band diagram was enlarged and examined at its central part for 6–7 eV interval (see Fig. 11). According to calculations [21,22] the maximum of valence band is situated in Z point and minima of conduction band are situated into the Brillouin zone center. From results of band structure calculations it is possible to mark Z, Г, P and V points. In these points vicinity maxima of valence bands and minima of conduction bands were observed at the same wavevector. Exactly in these points of Brillouin zone where selection rule is carried out electron transitions take place. For energy E4Eg in reflection spectra of E||c polarization the lowest energy maximum a1 was discovered. It is agreed with energy of direct electronic transition Г1(V1)-Г1(C1) defined from absorption spectra. For E⊥c polarization the lowest maximum e1 was observed at 2.407 eV and corresponded to Г2(V3)-Г1(C1) transitions. In this polarization almost at the same energy is observed e2 (2.543 eV) maximum which is most probably caused by Г1 (V4)-Г1(C1) transitions. Optical transitions a1, a2, e1 and e2 are observed in experimental reflection and ε2 and d2ε2/dE2 optical functions spectra are revealed in narrow energy interval. According to theoretical calculations [21,22,24,25] only in Brillouin zone center into the minimum of band gap two (three) degenerated band present which are split by crystalline field and spin-orbital interaction. Since the value of electron transitions splitting is very small Еgx┴–Еgx||(а1)¼35 meV, е1 (а2)–а1¼100 meV and е2–е1(а2)¼136 meV one can consider that а1, а2, е1, е2 features are caused by transition at k¼0 (Table 2). In covalent bond approach the value of spin-orbital splitting of valence band was estimated for cadmium phosphide Δso ¼0.06 eV. In the cases of singly and doubly ionized ions of Cd the spin-orbital splitting is equal to 0.15 and 0.30 eV, correspondingly. According to experimental dates the splitting of valence bands is equal to 35 meV (V2– V3) and 100–136 meV (V1–V2) due to crystalline field and spin-orbital interaction correspondingly. Next singularities a2–a8 (e3–e8) of reflection and optical functions spectra were found out in the energy interval 2.8–48– 4.2 eV. The nearest to Г(V1)–Г(C1) gap is a energy gap in Z point according to theoretical calculations of band structure. The easiest dispersion laws occur in Z and V points. They with and without taking into account spin–orbital interaction take the form Е(k,ε) ¼ Δ1 +ι(kх+kу). According to selection rules transition between states Z1,2-Z1+Z2, (Z1(s),Z4(s)), (Z2(s),Z3(s))-Z5+Z5, Z5(s)-(Z1(s) +Z4(s))+ (Z2(s) +Z3(s)) and states Z1,2-Z1,2 (Z1(s), Z4(s))-(Z2(s), Z3(s)), (Z2(s), Z3(s))-(Z1(s), Z4(s)) in polarization E⊥c and E||c are allowed, correspondingly [23]. In Z point of Brillouin zone according to theoretical calculations two degenerated valence bands and two conduction bands take place. Three valence bands appear in Z point in the case of splitting due to spin–orbital interaction and crystalline field. The lowest conduction band in Z point is split into two states Z1(C1) and Z2(C2) in the case of real crystals, which
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Table 2 Energies of electronic transitions in eV determined from measured reflection spectra and calculated d2ε2/dE2 spectra of CdP2 crystals. No. ind
Egx Egxdir (2 K) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14
R, eV E||c 2.1547 2.152 [2] 2.2727 2.267 2.404 2.825 3.020 3.179 3.455 4.005 4.265 4.541 4.758 4.895 5.409 5.605 6.076
d2ε2/dE2, eV
R, eV E⊥c
No. ind
2.271
2.834 3.045 3.194 3.483 4.012 4.265 4.555 4.736 4.866 5.395 5.598 6.068
Egx 2.153 [2] Egxdir (2 K) e1 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16
Fig. 12. Electron transitions into the structure of energy bangs of CdP2 crystal calculated in [24,25] and the fragment of Brillouin zone (insert) for crystals with fluorite symmetry.
means that, features of reflection and optical function spectra a3– a8 (e3–e8) were stipulated by transitions Z1(V1)-Z1(C1), Z1(V2)Z1(C1), Z1(V3)-Z1(C1) etc (Table 2). The next energy gap with higher energy is situated in P point of Brillouin zone. The energy dependence E(k, ε) in P point (point of localization of extrema valence and conduction band) without and with taking into account spin-orbital interaction has a next form Eðk; εÞ ¼ Δ1 þ ιkу þ mkx þ nkx kx þ pεzz
ð2Þ
Electron transitions between states P1,2-P1+P2, P1,2(s)P1(s)+P2(s) and P1,2-P2,1, P1,2(s)-P1,2(s) at polarization E⊥c and E|| c in the P point of Brillouin zone are allowed, correspondingly. Experimentally identified maxima e9–e12 and a9–a12 are caused probably by electron transitions Р1(V1)-Р1(С1), Р2(V2)-Р1(C1), Р1(V1)-Р2(С2) и Р2(V2)-Р2(С2). In V point of Brillouin zone weak extrema of valence and conduction bands at the same wavevector also take place. Dispersion laws in V point with and without spin–orbital interaction are simply [23] and looks like Е(k, ε)¼Δ1 +ι(kх+kу). According to selection rules transitions V1,3-V2 +V4, V2,4-V1 +V3, V1,3(s)V1(s) +V3(s), V2,4(s)-V2(s) +V4(s) are allowed at E⊥c polarization and transitions between states V1,3-V1,3, V2,4-V4,2, V1,4(s)-V4,1(s), V2,3(s)-V3,2(s) are allowed for E||c. Thus, features of reflection R, functions ε2 and d2ε2/dE2 spectra (e13, e14, a14–a16) are conditioned by transitions in V point of Brillouin zone, see Table 2.
d2ε2/dE2, eV
2.1547 2.3078 2.393 2.881 3.107 3.267 3.517 4.082
2.407 2.914 3.107 3.276 3.522 4.054
4.459 4.744 5.111 5.511 5.661 6.221 7.263 8.863
4.430 4.734 5.091 5.473 5.660 6.218 7.234 8.834
Model [21,22]
Model [24,25]
Z1 (V1)-Г1(C1) Г1 (V1)-Г1(C1) Г2(V2)-Г1(C1) Г2(V3)-Г1(C1) Z1(V1)-Z1(C1) Z2(V2)-Z1(C1) Z1(V3)-Z1(C1) Z1(V1)-Z2(C2) Z2(V2)-Z2(C2) Z1(V3)-Z2(C2) P1(V1)-P1(C1) P2(V2)-P1(C1) P1(V1)-P2(C2) P2(V2)-P2(C2) V2(V1)-V1(C1) V1(V2)-V1(C1) V-V, Z-Z, P-P
Г(V1)-Х(С1) Г1 (V1)-Г1(C1) Г2(V2)-Г1(C1) Г2(V3)-Г1(C1) X5(V1)-X1(C1) X5(V2)-X1(C1) X5(V3)-X1(C1) X5(V1)-X1(C2) X5(V2)-X1(C2) X5(V3)-X1(C2) L3(V1)-L1(C1) L3(V2)-L1(C1) L3(V3)-L1(C1) Г1(V1)-Г25(C2) Г2(V2)-Г25(C2) Г1(V3)-Г25(C2) L or Г
As mentioned above according to calculations of works [24,25] two (three) degenerated bands into the minimum of band-gap of Brillouin zone center are present. These bands were split by crystalline field and spin–orbital interaction in the case of real crystal. Based on calculations [24,25], experimentally observed electron transitions in points of Brillouin zone for which maxima of valence bands and minima of conduction bands have the same wavevector k were scrutinized (see Fig. 12). Experimentally received energy gaps were analyzed within the limits of the energy increasing. They localized for band model in order of increasing of interband gaps too. At the same time the amount of bands in every points of Briillouin zone for the used model was examined by taking into account the splitting of degenerated bands due to spin-orbital interaction and influence of crystalline field. These were indicated by experimentally discovered electron transitions which group in narrow energy region. Also these were shown by the energy difference of same maxima aj(ej). At the same time in band structures received by theoretical calculations in works [21,22,24,25] energy distance between bands C–C and V–V into the points of extrema localization are much more (1 eV) than the ones between energies of maxima aj(ej). 4. Conclusions The maximum of transmission at λо∼896 nm is observed for CdP2 crystals in crossed polarizers, refractive indexes nk||с and nk||у which intersect at this wavelength. In high wavelength part of λ0 the value of refractive index nk||с is more than in the case of another polarization nk||у. The opposite dependence is observed for short wavelength region. Indirect transitions in excitonic region Еgx are nonpolarized and correspond to transition from band Z1 to band Г1. Minima direct transitions Г1-Г1 (E||c) and Г2-Г1 (E⊥c) are split and have a temperature coefficient of energy gaps shifting. Features of optical functions (n, k, ε1, ε2, d2ε1/dE2 and d2ε2/dE2) were determined for energies from 1.5 to 10 eV and interpreted on the basis of existing theoretical calculations of band structure. References [1] V.B. Lazarev, V.J. Shevchenko, J.H. Greenberg, V.V. Sobolev, Compound semiconductors of the AIIBV group, Science, Мoscow 1978 (in Russian). [2] N.N. Syrbu, Optoelectronic properties of II–V compounds, Stiinta, Kishinev 1983 (in Russian). [3] N.N. Syrbu, I.G. Stamov, A.I. Kamertsel, Sov. Phys. Semicond. 26 (1992) 665.
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[4] V.V. Borshch, V.A. Gnatyuk, S.A. Kovalenko, M.G. Kuzmenko, R.V. Yaremko, in: Proceedings of SPIE 5024, Selected Papers on Optics and Photonics: optical Diagnostics of Materials and Devices for Opto-, Micro-, and Quantum Electronics, 1 (2003) 112. [5] N.N. Syrbu, A.I. Kamertsel, I.G. Stamov, Sov. Phys. Semicond. 24 (1992) 376. [6] K.B. Aleinicova, A.I. Kozlov, S.G. Kozlova, V.V. Sobolev, Phys. Solid State 44 (2002) 1257. [7] I.G. Stamov, D.V. Tkacenco, Semiconductors 42 (2008) 1. [8] I.G. Stamov, D.V. Tkacenco, Semiconductors 40 (2006) 1165. [9] N. Sobotta, H. Neumann, V. Riede, N. Syrbu, Solid State Commun. 48 (1983) 297. [10] I.I. Patscun, I.A. Sliukhina, Semicond. Phys. Quantum Electron. Optoelectron. 7 (2005) 31. [11] V. Yu. Rud’, Yu.V. Rud’, I.V. Bodnar’, Tech. Phys. 55 (2010) 517. [12] V.A. Gnatyuk, V.V. Borshch, M.G. Kuzmenko, O.S. Gorodnychenko, S.O. Yuryev, Phys. Chem. Solid State 5 (2004) 256. [13] V.M. Trukhan, A.U. Sheleg, I.F. Fekeshgazi, Photoelectronics 13 (2004) 15. [14] I.V. Bodnari, M.A. Osipova, V.Y.u. Rud, Y.u.V. Rud, B.K.h. Bairamov, J. Appl. Spectrosc. 77 (2010) 146.
19
[15] V.M. Trukhan, A.U. Sheleg and all. Patent Belorus No. 12388, 2009. [16] V.M. Trukhan, L.E. Soshnikov, S.E. Marenkin, T.V. Haliakevich, Inorg. Mater. 41 (2005) 901. [17] A.U. Sheleg, A.A. Kutas, V.N. Yakimovich, V.G. Fedotov, D.N. Karlikov, L.M. Gorynia, Patent SU, No. 875959, 1981. [18] N.N. Syrbu, I.G. Stamov, A.G. Umanetc, Patent SU, No. 816345, (1980), Patent SU, No. 865040, 1981. [19] A.A. Andreev, B.T. Meleh, V.M. Truhan, Patent SU, No. 621254, 1978. [20] V.P. Novikov, A.U. Sheleg, V.M. Truhan, V.G. Fedotov, A.G. Byckov, D.N. Karlikov, L.M. Gorynea, Patent SU, No. 917004, 1981. [21] I. Yu., A.S. Polygalov, V.E. Poplavnoi, Tupitsyn, Izvestia Vuzov SSSR, Seria Fizica 3 (1981) 123. [22] V.E. Tupitsyn, Y.u.I. Polygalov, A.S. Poplavnoi, Fiz. Tekh. Poluprovodn. 15 (1981) 2414. [23] V.V. Sobolev, N.N. Syrbu, T.M. Sushkevich, Phys. Status. Solidi (B) 43 (1971) 73. [24] P.J. Lin-Chung, Phys. Rev. 188 (1969) 1272. [25] P.J. Lin-Chung, Phys. Status. Solidi (B) 47 (1971) 33. [26] D.O. Scanion, A. Walsh, Appl. Phys. Lett. 100 (2012) 251911.
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Physica B journal homepage: www.elsevier.com/locate/physb
Birefringence and band structure of CdP2 crystals S.I. Beril a, I.G. Stamov a, N.N. Syrbu b,n, V.V. Zalamai c a b c
Tiraspol State Corporative University, Yablocikin Street 5, 2069 Tiraspol, Republic of Moldova Technical University of Moldova, 168 Stefan cel Mare Avenue, 2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Street, 2028 Chisinau, Republic of Moldova
art ic l e i nf o
a b s t r a c t
Article history: Received 12 February 2013 Received in revised form 4 April 2013 Accepted 6 April 2013 Available online 20 April 2013
The spatial dispersion in CdP2 crystals was investigated. The dispersion is positive (nk||с 4nk||у) at λ4 λ0 and negative (nk||с o nk||у) at λ oλ0. CdP2 crystals are isotropic for wavelength λо ¼ 896 nm. Indirect transitions in excitonic region Еgx are nonpolarized due to one pair of bands. Minimal direct energy intervals correspond to transitions Г1-Г1 for Е||с and Г2-Г1 for Е⊥с. The temperature coefficient of energy gap sifting in the case of temperature changing between 2 and 4.2 K equals to 10.6 meV/K and 3.2 mev/K for Г1-Г1 and Г2-Г1 band gap correspondingly. Reflectivity spectra were measured for energy interval 1.5–10 eV and optical functions (n, k, ε1, ε2, d2ε1/dE2 and d2ε2/dE2) were calculated by using Kramers–Kronig analyses. All features were interpreted as optical transitions on the basis of both theoretical calculations of band structure. & 2013 Elsevier B.V. All rights reserved.
Keywords: Semiconductor compound Optical absorption and reflection spectra Kramers–Kronig relations Optical constants Excitons Band structure
1. Introduction The cadmium diphosphide is a wide-gap AIIBV semiconductor and possesses the anisotropy of optical properties with natural gyrotropy [1–3]. The optical spectra of photoluminescence and photoconductivity, transmission and reflection for the region of electronic transitions and vibrational modes are polarized [3–12]. Quantum electronic and nonlinear optic devices were developed on the basis of CdP2 crystals whose operating principle was based on gyrotropy and nonlinear polarizability of the crystal. Values of nonlinear polarizability and gyrotropy of cadmium diphosphide are higher than other crystals and they provide effective stabilization of radiation field in the space (lateral section) and in the time simultaneously [13]. Stabilizers of emission field and extenders of lasers impulse duration on the basis of CdP2 were developed. The low thermal-conductivity (10 W/mK) of CdP2 crystals was used for producing laser beam deflectors with thermal induced gradient of refractive index. Advantage of deflectors is a possibility of their usage in visible and infrared regions [14,15]. The linear dependence of polarization plane of plane-polarized light on magnetic field induction was found in cadmium diphosphide and a possibility of producing on the basis of CdP2 of magnetooptical modulators and magneto-optical sensible elements of measuring of magnetic field induction was shown [16,17].
n
Corresponding author. Tel.:+373 22 237508. E-mail address: [email protected] (N.N. Syrbu).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.04.029
The cadmium diphosphide has a high photosensitivity and intense luminescence [3–5]. The technology of p–n junction producing surface-barrier diodes was developed on the basis of CdP2 crystals [2,3,5,7,8]. CdP2 based photoresistors have small relaxation time and are suitable for registration of pulse radiation with duration of 1 ns. Photoelectron emitters, photodiodes, electrical switchers and stabilitrons [18–20] were developed on the basis of CdP2 monocrystals. CdP2 crystals change the optical activity with temperature change at the same time the linear temperature dependence of rotatory power of polarization plane which remains invariable. Temperature sensors were developed due to this effect [20]. Last year the level of technology allows to grow more perfect and high quality crystals. In this work properties of birefraction, luminescence, and absorption in the region of band gap minimum were investigated for perfect crystals and new experimental results were received. Reflection spectra were researched for wide energy region (1–10 eV), calculated using Kramers–Kronig relations and discussed based on two variants of band structure theoretical calculations. 2. Experiment The process of diphosphide monocrystal growing from precursors Cd and P had two stages. The purifications initial substances and synthesis of CdP2 chemical solution were carried out during the first phase. The synthesis was executed in quartz ampoules which were deposited in high pressure (40 atm)
S.I. Beril et al. / Physica B 422 (2013) 12–19
chamber. Up to 300–500 g of material was synthesized simultaneously. During second stage the synthesized substance was treated in vacuum for receiving monocrystals. The maximal size of ingots of CdS2 grown from gas phase achieved 15 15 30 mm. Some single crystals had clear crystallographic shape and did not need subsequent treatment and orientation according to crystallographic axes. Monocrystals were cleaved out perpendicular to caxis. Measurements were executed for monocrystals grown along c-axis as a prism or plate with smooth nontreated surfaces. Measurements at E||c and Е⊥с, k||у polarizations were carried out on nontreated smooth surface (1 0 1). In the case of Е⊥с and k||с polarization samples cleaved out perpendicular to c-axis from the same ingot were used. Absorption spectra at 2 K and 4.2 K were measured in glass cryostat with liquid helium by means of spectrometer DFS-24 with resolution 5 Å/mm. The spectral width of spectrometer slit during the measurements of absorption spectra was less than 0.1 Å. Optical spectra of absorption and luminescence at 9 K were measured in cryostat LTS-32C330 Workhorse on spectrometer DFS-32 with the same value of spectral slit of spectrometer. Reflection spectra were measured by spectrometer Specord M40 at 77 K in energy range of 1–6 eV.
3. Results and discussion 3.1. Birefringence of CdP2 crystals The spectral dependence of transmission coefficient (T) of CdP2 crystals that deposit between two crossed polarizers is shown in Fig. 1. The position of polarizers and crystal axes orientation during measurements can be seen in inset. It was found that the maximum at wavelength 654 nm in transmission spectra of crystals with thickness d ¼ 1117 mm deposed between crossed polarizers (Fig. 1, curve a). Additional peaks appear at short and long wavelength parts of spectra by changing of crystal thickness to 739 and 470 mm, but at the same time the central line at 654 nm
Fig. 1. The spectral dependence of transmission (T) of CdP2 crystals with thicknesses 1.117 mm (a), 0.73 mm (b) and 0.47 mm (c) in crossed polarizers. The inset shows the orientation of crystal axes.
13
remain in spectra (see Fig. 1, curves b and c). The wavelength 654 nm is isotropic wavelength (λ0) for CdP2 crystals. The crystal does not recognize a polarization of light waves in this wavelength. The Fig. 1 shows the spectral dependence of transmission T in diapason from 0.6 to 2.5 mm for CdP2 crystals with thickness 1.117 mm (a), 0.73 mm (b) and 0.47 mm (c) deposed between crossed polarizers. The clear interference with lines intensifying in short wavelength part of spectra is observed. The difference between refractive indexes Δn ¼no−ne was defined by these dates. The curve Δn intersects the wavelength axis at isotropic wavelength λо (654 nm). This difference Δn¼ no−ne achieves minimal value (−0.3) at 2.0 mm wavelength. Reflectivity spectra for short wavelength part from isotropic wavelength were measured and refraction indexes for different polarizations of light wavelength E and direction of wavevector k were calculated by Kramers–Kronig (KK) relations (see Fig. 2b). Measurements of reflectivity spectra were carried out for two surfaces of crystal (1 1 0) and (1 0 1) almost at angles close to normal (Fig. 1). For determining parameters in Е||с, k||y, Е⊥с, k||y polarization light beams were directed on sample surface ac(xy). The electrical vector of light wave allocated in the crystallographic plane ac(xy) and the wavevector k was perpendicular to axis c (k⊥c). For polarizations Е||y, k||c, Е||x, k||c light rays were directed on surface ab(xy) parallel to the optical axis c. At sample living the plane of vibrations of electrical vector turns on angle (φ¼ ρd, where ρ—specific turn of vibrations plane) due to natural optical activity at low intensity of light. Spectral dependences of refractive indexes for polarizations Е|| с, k||y, Е⊥с, k||y и Е||y, k||c, Е||x, k||c calculated from reflectivity spectra by Kramers-Kronig relations were shown in Fig. 2. Refractive indexes for polarizations Е||y, k||c, Е||x, k||c change almost
Fig. 2. (A) Fragments of interference in transmission spectrum (T) of crystal CdP2 in crossed polarizers and spectral dependence Δn ¼no–ne, and (B) spectral dependences of refractive indexes no, ne и Δn¼ no–ne determined by Kramers–Kronig analyses.
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parallely. Also in polarization Е||с, k||y, Е⊥с, k||y the character of changing of refractive indexes is parallel, but they cross at energy 1.879 eV (896 nm). This cross-point is a isotropic point with wavelength λ0 for CdP2 crystals. The transmission maximum at this wavelength in transmission spectra of crystals at crossed polarizers is observed (Fig. 1, curve T). The value of refractive index nk||с is more than refractive index in polarization nk||у in longwavelength part of λ0, it means that dispersion is positive. The opposite dependence observes in short-wavelength region—the dispersion is negative. Such crystal represents a phase plate in which two light waves propagate with rates Vx ¼c/nk||с and Vy/nk||у. The spectral characteristic of the difference of refractive indexes Δn ¼no−ne determined by KK analyses and interference match and both curves have a zero value at 654 nm wavelength. At high intensity of radiation the additional turn of vibrational plane on angle γЕ02, whose value is proportional to light intensity, is observed [13]. In work [13] the efficiency of induced gyration by the dependence changing of transmission intensity of system polarizer-sample-analyzer was estimated. Since measured crystals take place linear and nonlinear losses of radiation thus the intensity of light after analyzer was determined experimentally. It was shown [13] that the dependence of specific value of angle φ/ d induced by nonlinear rotation of polarization plane of light in CdP2 crystals has been proportional to the intensity of light J0. The edge absorption in CdP2 crystals at room temperature has a gentle increase of absorption coefficient for both polarizations (Е|| с, Е⊥с). The temperature decrease causes a shift of absorption edge in short-wavelength region. At low temperatures in region of small absorption coefficients in spectra appear features in Е⊥с and Е||c polarizations which correspond to indirect transitions with phonon emission. These features one can see clearly in spectra d2K/ dE2; they have been marked in Fig. 3 as ξ1–ξ4. The position of ξ1– ξ4 maxima almost coincide in both polarizations. The threshold value of nonphonon indirect transition in excitonic band Еgx is equal to ξ1–Еf, where Еf is energy of phonon participated in
indirect transitions. The minimal energy of phonons 5.4 meV (43.5 cm−1), 9.5 meV (77 cm−1), 9.72 meV (78.4 cm−1), 9.97 meV (80.4 cm−1) by Raman analyses of CdP2 crystals was determined. It turned out almost all vibrational modes in high energy part of spectra. Indirect phonon-aided transitions begin with participation of low-energy optical phonon with energy 5.4 meV (43.5 cm−1). In this case the indirect energy gap in excitonic band Еgx equals to ξ1 (2.16 meV)–Еf (5.4 meV) ¼2.1547 eV. The threshold value of nonphonon indirect transition to excitonic band coincides for polarizations Е⊥с and Е||c and equals to 2.1547 eV with accuracy about 0.0005 eV. CdP2 crystals have a structure described by space group D84 (D44 ). Eight formulaic units compose a unit cell i.e. 24 atoms and amount of phonons in usual case is equal to 72. The presence of such amount of vibrational modes allows observing multiphonon emission of free excitons and indirect transitions with multiphonon absorption and emission. Threshold values of indirect excitonic transitions and phonon energy are presented in work [2]. Intensive lines Еgx1 and Egx0 (2.1533 eV and 2.1547 eV) were discovered in luminescence spectra of CdP2 crystals measured at 9 K and excited by 4765 Å lines of Ar laser. Lines a1–a17 were observed in long-wavelength part from these lines (see Fig. 4). The energy of maximum Egx0 (2.1547 eV) matches with the energy of indirect transition in excitonic band. Exactly this value was received from spectra of modulated absorption d2K/dE2. The energy distance between emission lines Еgx1 and Egx0 equals 1.4 meV. This splitting causes by the splitting of excitonic states due to the exchange interaction. Maxima a1–a17 correspond to phonon emission at free indirect exciton annihilation. Table 1 shows an energy position of discovered luminescence lines and participated phonons. Direct transitions are observed in energy region higher than energy of indirect transition (2.1534 eV). Spectral characteristics of edge absorption in polarizations E||c and E⊥c at temperatures 300, 80, 4.2 and 2 K do not coincide (Fig. 5). Edges of absorption intersect at the wavelength λ0 for the diapason of absorption coefficients 200–400 cm−1. The edge of absorption for E⊥c is shifted to low-energy part in comparison with E||c polarization for all high values of absorption coefficients. The sharp increasing of absorption coefficient for this diapason is caused by direct allowed transitions. Calculated values of absorption coefficient coincide with experiment at appropriate parameters: Еgx ¼ 2.2727 eV and А¼3.04 104 eV1/2/сm and Еgx ¼ 2.3078 eV and А¼2.8 104 eV1/2/сm at temperature 2 K and polarization Е⊥с and E||c correspondingly, where A¼
2e2 ð2mnÞ3=2 P CV j2 m20 cℏ2 n
ð1Þ
Received results show that absorption curves are shifted on 32.1 meV in region of direct allowed transitions at 2 K for E||c and
Fig. 3. Absorption spectra K and their energy derivatives d2K/dE2 of CdP2 crystals measured at 9 K.
Fig. 4. Luminescence spectra of CdP2 crystals measured at 9 K and excited by 476.5 nm Ar laser line.
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Table 1 Maxima of emission spectra of CdP2 crystals at 9 K. No. ai
ai
Egx (2.1534)–ai, meV
IR, R, meV (Г2, А2) [4], [2]*
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17
2.1440 2.1419 2.1386 2.1356 2.1274 2.1253 2.1209 2.1190 2.1150 2.1106 2.1073 2.1049 2.1002 2.0961 2.0945 2.0922 2.0889
9.4 11.4 14.7 17.7 25.9 28.0 32.4 34.3 38.4 42.7 46.0 48.4 53.1 57.2 58.8 61.1 64.5
9.5* 11.4* 14.3 17.5* 26.16 11.4+17.7 14.7+17.7 28.0+6.2* 38.1 28+14.7 28+17.7 9.4+39.3 55.5
Г5 [2]*
26.2
38.1
55.4 57.5
15
of valence and conduction bands is observed i.e. then k ¼0. The band symmetry according to selection rules for direct electronic transition was established in Г point [2,23]. In case of E⊥c polarization selection rules allow transition of symmetry Г1,2,3,4-Г5, Г5-Г1+Г2+Г3+Г4 and transitions of symmetry Г1,2(s)-Г1(s)+Г2(s) which permit taking into account a spinorbital interaction. The minimal direct energy interval (2.2727 eV) is therefore observed at E⊥c polarization due to transitions Г1-Г1 or Г2- Г2. For E||c polarization transitions between bands with symmetries Г1,2,3,4-Г2,1,4,3, Г5-Г5 and Г1,2(s)-Г1,2(s) are allowed by selection rules. The minimum of conduction band has symmetry Г1 according to all calculated models of band structure. Therefore the upper valence band V1 has a symmetry Г1 in point Г and V2 band has Г2 symmetry (see Fig. 4). The top valence band V1 in point Z has symmetry Z1. Indirect transitions from band Z1(V1) to band Г1(С1) are allowed for both polarizations but they are not observed experimentally.
58.0 38.4+25.9
3.2. Structure of energy bands for CdP2-D48 crystals Reflectivity spectra in the depth of fundamental absorption of CdP2-D48 crystals at temperature 77 K for polarizations E||c and E⊥c for energy interval 1–6 eV and at 300 K for energies 6–10 eV were measured (see Fig. 6A). The absorption coefficient K was calculated for both polarizations from measured reflection spectra by using Kramers–Kronig equations (Fig. 6B). At intrinsic zone (E 1.5–10 eV) in reflection spectra maxima a1–a14 for E||c polarization and e1–e16 for E⊥c polarization are
Fig. 5. Absorption spectra of CdP2 crystals with thickness 7.3 mm in polarized light measured at 4.2 and 2 K temperatures.
E⊥c polarizations, thus direct minimal transitions come from two valence bands. These transitions are allowed in different polarizations by selection rules. The temperature increasing from 2 to 4.2 K causes the long-wavelength shift of absorption edge. The temperature coefficient of energy interval shift ΔE=ΔT stipulated by allowed direct transitions is equal to 10.6 meV/K and 3.2 meV/K for polarization E||c and E⊥c correspondingly at temperature changing from 2 to 4.2 K. Different coefficients of edge absorption temperature shift for E||c and E⊥c polarizations argue whether direct transitions come from two bands V1 and V2. Received experimental results give evidence that band gap Eg1ind corresponds to indirect transitions. According to calculations of band structure of CdP2 crystals the maximum of valence band is localized in point Z of Brillouin zone [21,22]. Therefore indirect transitions occur between extremum Г–Z (Z–Г) in E||c and E⊥c polarizations. The fragment of band structure in band minimum is shown in inset of Fig.5. Indirect transitions are nonpolarized since these Г–Z transitions with phonon participation are allowed by selection rules for both polarizations [2]. Direct transitions take place without phonon and thus they are polarized [2,23]. In the centrum of Brillouin zone without taking into account spin-orbital interaction occur next dispersion laws Е(k,ε) ¼ αkZ2+bk┴2+eεzz+dε┴ ¼Δ1; (k┴2 ¼kх2+kу2, ε┴ ¼εхх+εуу) and with taking into account spin–orbital interaction Е(k,ε) ¼Δ1+ι(kх+kу)7 (m2kz2)1/2 [23]. Dispersion laws are presented for values of wavevector where according to theoretical calculations extremum
Fig. 6. Reflection spectra of CdP2 crystals measured in polarized light (1–6 eV) at 77 K and in nonpolarized light (6–10 eV) at 300 K. The inset shows a model of band structure for band-gap minimum.
16
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observed. The absorption coefficient has a high value in E||c polarization for all measured diapason and changes in limit of (2.5–5.5) 105 cm−1. Fig. 6 shows that the absorption for intrinsic zone in polarization E||c is less than in polarization E⊥с. Optical functions n, k, (Fig. 7) ε1, ε2, d2ε1/dE2 and d2ε2/dE2 (Figs. 8 and 9) were calculated for all measured diapason in both polarizations by Kramers–Kronig analyses. Maxima of reflectivity spectra and optical factions (n, k, ε1, ε2 and d2ε2/dE2) were clearly recognized at 77 K and also at 300 K (see Fig. 10). Features observed in reflection spectra and optical functions are caused by direct optical transitions in actual points in Brillouin zone. The intensive picks observed at 77 K were detected also at room temperature in reflection spectra and thus in optical function
Fig. 9. The spectral dependence of optical functions ε2, d2ε2/dE2 and ε1, d2ε1/dE2 of CdP2 crystals for E||c measured at 77 K. Fig. 7. The spectral dependence of optical functions n and k calculated by Kraners– Kronig analyses from reflectivity spectra of CdP2 crystals in E||c and E⊥c polarizations measured at 77 K.
Fig. 8. The spectral dependence of optical functions ε1, d2ε1/dE2, and ε2, d2ε2/dE2 of CdP2 in E⊥c polarization measured at 77 K.
spectra. These features are separated in to four groups of maxima Еg–а2 (е2), а3 (е3)–а8 (е8), а9 (е9)–а12 (е12) and high-energy group а13 (е13)–а16. These groups of transitions come from extremum of valence bands to extremum of conduction band at the same value of wavevector k. The Brillouin zone is a rectangular prism with 14 singular points [23]. Group-theoretical calculations of dispersion laws in actual points of Brillouin zone and calculation of selection rules of CdP2 for all points of Brillouin zone with and without taking into account the spin-orbital interaction were carried out [2,23]. Actual points of Brillouin zone (points of zero slope of energy), dispersion laws and selection rules in these points were defined. The zero energy slope realizes by two of three directions kx, ky, kz for Г, Z, М, R, Σ, Λ, N, X, R, А points. For all three directions the zero slope of energy realizes in points Г, М, R without taking into account a spin-orbital interaction. In the case of taking a spin into account the zero slope of energy for all directions fulfils in Г point. Theoretical calculation of band structure show extrema exactly in Г, Р, Z and V points of Brillouin zone [21,22]. The ratios of dispersion analyses show that they have the simplest view in Г, X, P, N, Z points. Transitions in R, X, Y, Т, S and Σ points are not polarized according to selection rules. Selection rules in centrum of Brillouin zone indicate that direct transitions have a different energy interval for each polarization that is polarized. Theoretical calculations of band structure at a wide energy diapason into different points of Brillouin zone for cadmium phosphide with D2h20 symmetry were executed by pseudopotential
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17
Fig. 11. Electronic transitions into the structure of energy bands of CdP2 crystals [23,24] and Brillouin zone (insert) of crystals with D48 symmetry.
Fig. 10. The spectral dependence of optical function ε2 and its derivation d2ε2/dE2 of CdP2 crystals measured at 300 K for E⊥c (A) and E||c (B) polarizations.
method [24,25]. The crystal lattice of Cd3P2 was considered as an analog of lattice with structure of fluorite during theoretical calculations. Cd atoms are introduced as vacancies periodically deposed in lattice of crystal. The maximum of valence band and the minimum of conduction band are found in Г point of Brillouin zone center. The spin–orbital interaction and another relativistic effect are not taken into account during calculation. The maximum of valence band in Г point is triply degenerated. Taking into account the crystal field potential and spin-orbital interaction lead to the valence band splitting in the case of real crystals. The maximum of valence band Г15 splits in three bands due to degeneration removing. Valence bands have maxima and conduction bands have minima at the same wavevector k in Г, Х and L points. Theoretical calculations of band structure of CdP2 and ZnP2 crystals executed by pseudopotential method in Г, Z, V, U, R, А, Р, М and Λ points of symmetry at wide energy diapason (25 eV) were described in works [21,22]. A huge amount of bands with flat character was received. The lowest conduction band as well as calculations presented in [24,25] is single and its minimum situated in Γ point. Valence bands are also flat. The work [26] shows calculations of band structure of ordered chalcopyrite ZnSnP2 and disordered structure of sphalerite Zn(Sn)P2 in Г, Х, Р, N and Z points of Brillouin zone. The lower conduction band is formed by s-states of Zn atoms and the upper valence band is formed by p-states of phosphor atoms. Received band diagrams of Zn(Sn)P2 [26] and Zn3P2 [24,25] crystals is similar and do not contradict with the results for CdP2, ZnP2 [21,22]. In this work experimental results are discussed in framework of band model [21,22,24,25] subject to dispersion laws into the points of extrema localization. In the structure of energy bands presented in [21,22]
it is a bit difficult to recognize values of wavevector where maxima of valence band and minima of conduction band coincide. For comparison of experimental and theoretical energy gaps the theoretical band diagram was enlarged and examined at its central part for 6–7 eV interval (see Fig. 11). According to calculations [21,22] the maximum of valence band is situated in Z point and minima of conduction band are situated into the Brillouin zone center. From results of band structure calculations it is possible to mark Z, Г, P and V points. In these points vicinity maxima of valence bands and minima of conduction bands were observed at the same wavevector. Exactly in these points of Brillouin zone where selection rule is carried out electron transitions take place. For energy E4Eg in reflection spectra of E||c polarization the lowest energy maximum a1 was discovered. It is agreed with energy of direct electronic transition Г1(V1)-Г1(C1) defined from absorption spectra. For E⊥c polarization the lowest maximum e1 was observed at 2.407 eV and corresponded to Г2(V3)-Г1(C1) transitions. In this polarization almost at the same energy is observed e2 (2.543 eV) maximum which is most probably caused by Г1 (V4)-Г1(C1) transitions. Optical transitions a1, a2, e1 and e2 are observed in experimental reflection and ε2 and d2ε2/dE2 optical functions spectra are revealed in narrow energy interval. According to theoretical calculations [21,22,24,25] only in Brillouin zone center into the minimum of band gap two (three) degenerated band present which are split by crystalline field and spin-orbital interaction. Since the value of electron transitions splitting is very small Еgx┴–Еgx||(а1)¼35 meV, е1 (а2)–а1¼100 meV and е2–е1(а2)¼136 meV one can consider that а1, а2, е1, е2 features are caused by transition at k¼0 (Table 2). In covalent bond approach the value of spin-orbital splitting of valence band was estimated for cadmium phosphide Δso ¼0.06 eV. In the cases of singly and doubly ionized ions of Cd the spin-orbital splitting is equal to 0.15 and 0.30 eV, correspondingly. According to experimental dates the splitting of valence bands is equal to 35 meV (V2– V3) and 100–136 meV (V1–V2) due to crystalline field and spin-orbital interaction correspondingly. Next singularities a2–a8 (e3–e8) of reflection and optical functions spectra were found out in the energy interval 2.8–48– 4.2 eV. The nearest to Г(V1)–Г(C1) gap is a energy gap in Z point according to theoretical calculations of band structure. The easiest dispersion laws occur in Z and V points. They with and without taking into account spin–orbital interaction take the form Е(k,ε) ¼ Δ1 +ι(kх+kу). According to selection rules transition between states Z1,2-Z1+Z2, (Z1(s),Z4(s)), (Z2(s),Z3(s))-Z5+Z5, Z5(s)-(Z1(s) +Z4(s))+ (Z2(s) +Z3(s)) and states Z1,2-Z1,2 (Z1(s), Z4(s))-(Z2(s), Z3(s)), (Z2(s), Z3(s))-(Z1(s), Z4(s)) in polarization E⊥c and E||c are allowed, correspondingly [23]. In Z point of Brillouin zone according to theoretical calculations two degenerated valence bands and two conduction bands take place. Three valence bands appear in Z point in the case of splitting due to spin–orbital interaction and crystalline field. The lowest conduction band in Z point is split into two states Z1(C1) and Z2(C2) in the case of real crystals, which
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S.I. Beril et al. / Physica B 422 (2013) 12–19
Table 2 Energies of electronic transitions in eV determined from measured reflection spectra and calculated d2ε2/dE2 spectra of CdP2 crystals. No. ind
Egx Egxdir (2 K) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14
R, eV E||c 2.1547 2.152 [2] 2.2727 2.267 2.404 2.825 3.020 3.179 3.455 4.005 4.265 4.541 4.758 4.895 5.409 5.605 6.076
d2ε2/dE2, eV
R, eV E⊥c
No. ind
2.271
2.834 3.045 3.194 3.483 4.012 4.265 4.555 4.736 4.866 5.395 5.598 6.068
Egx 2.153 [2] Egxdir (2 K) e1 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16
Fig. 12. Electron transitions into the structure of energy bangs of CdP2 crystal calculated in [24,25] and the fragment of Brillouin zone (insert) for crystals with fluorite symmetry.
means that, features of reflection and optical function spectra a3– a8 (e3–e8) were stipulated by transitions Z1(V1)-Z1(C1), Z1(V2)Z1(C1), Z1(V3)-Z1(C1) etc (Table 2). The next energy gap with higher energy is situated in P point of Brillouin zone. The energy dependence E(k, ε) in P point (point of localization of extrema valence and conduction band) without and with taking into account spin-orbital interaction has a next form Eðk; εÞ ¼ Δ1 þ ιkу þ mkx þ nkx kx þ pεzz
ð2Þ
Electron transitions between states P1,2-P1+P2, P1,2(s)P1(s)+P2(s) and P1,2-P2,1, P1,2(s)-P1,2(s) at polarization E⊥c and E|| c in the P point of Brillouin zone are allowed, correspondingly. Experimentally identified maxima e9–e12 and a9–a12 are caused probably by electron transitions Р1(V1)-Р1(С1), Р2(V2)-Р1(C1), Р1(V1)-Р2(С2) и Р2(V2)-Р2(С2). In V point of Brillouin zone weak extrema of valence and conduction bands at the same wavevector also take place. Dispersion laws in V point with and without spin–orbital interaction are simply [23] and looks like Е(k, ε)¼Δ1 +ι(kх+kу). According to selection rules transitions V1,3-V2 +V4, V2,4-V1 +V3, V1,3(s)V1(s) +V3(s), V2,4(s)-V2(s) +V4(s) are allowed at E⊥c polarization and transitions between states V1,3-V1,3, V2,4-V4,2, V1,4(s)-V4,1(s), V2,3(s)-V3,2(s) are allowed for E||c. Thus, features of reflection R, functions ε2 and d2ε2/dE2 spectra (e13, e14, a14–a16) are conditioned by transitions in V point of Brillouin zone, see Table 2.
d2ε2/dE2, eV
2.1547 2.3078 2.393 2.881 3.107 3.267 3.517 4.082
2.407 2.914 3.107 3.276 3.522 4.054
4.459 4.744 5.111 5.511 5.661 6.221 7.263 8.863
4.430 4.734 5.091 5.473 5.660 6.218 7.234 8.834
Model [21,22]
Model [24,25]
Z1 (V1)-Г1(C1) Г1 (V1)-Г1(C1) Г2(V2)-Г1(C1) Г2(V3)-Г1(C1) Z1(V1)-Z1(C1) Z2(V2)-Z1(C1) Z1(V3)-Z1(C1) Z1(V1)-Z2(C2) Z2(V2)-Z2(C2) Z1(V3)-Z2(C2) P1(V1)-P1(C1) P2(V2)-P1(C1) P1(V1)-P2(C2) P2(V2)-P2(C2) V2(V1)-V1(C1) V1(V2)-V1(C1) V-V, Z-Z, P-P
Г(V1)-Х(С1) Г1 (V1)-Г1(C1) Г2(V2)-Г1(C1) Г2(V3)-Г1(C1) X5(V1)-X1(C1) X5(V2)-X1(C1) X5(V3)-X1(C1) X5(V1)-X1(C2) X5(V2)-X1(C2) X5(V3)-X1(C2) L3(V1)-L1(C1) L3(V2)-L1(C1) L3(V3)-L1(C1) Г1(V1)-Г25(C2) Г2(V2)-Г25(C2) Г1(V3)-Г25(C2) L or Г
As mentioned above according to calculations of works [24,25] two (three) degenerated bands into the minimum of band-gap of Brillouin zone center are present. These bands were split by crystalline field and spin–orbital interaction in the case of real crystal. Based on calculations [24,25], experimentally observed electron transitions in points of Brillouin zone for which maxima of valence bands and minima of conduction bands have the same wavevector k were scrutinized (see Fig. 12). Experimentally received energy gaps were analyzed within the limits of the energy increasing. They localized for band model in order of increasing of interband gaps too. At the same time the amount of bands in every points of Briillouin zone for the used model was examined by taking into account the splitting of degenerated bands due to spin-orbital interaction and influence of crystalline field. These were indicated by experimentally discovered electron transitions which group in narrow energy region. Also these were shown by the energy difference of same maxima aj(ej). At the same time in band structures received by theoretical calculations in works [21,22,24,25] energy distance between bands C–C and V–V into the points of extrema localization are much more (1 eV) than the ones between energies of maxima aj(ej). 4. Conclusions The maximum of transmission at λо∼896 nm is observed for CdP2 crystals in crossed polarizers, refractive indexes nk||с and nk||у which intersect at this wavelength. In high wavelength part of λ0 the value of refractive index nk||с is more than in the case of another polarization nk||у. The opposite dependence is observed for short wavelength region. Indirect transitions in excitonic region Еgx are nonpolarized and correspond to transition from band Z1 to band Г1. Minima direct transitions Г1-Г1 (E||c) and Г2-Г1 (E⊥c) are split and have a temperature coefficient of energy gaps shifting. Features of optical functions (n, k, ε1, ε2, d2ε1/dE2 and d2ε2/dE2) were determined for energies from 1.5 to 10 eV and interpreted on the basis of existing theoretical calculations of band structure. References [1] V.B. Lazarev, V.J. Shevchenko, J.H. Greenberg, V.V. Sobolev, Compound semiconductors of the AIIBV group, Science, Мoscow 1978 (in Russian). [2] N.N. Syrbu, Optoelectronic properties of II–V compounds, Stiinta, Kishinev 1983 (in Russian). [3] N.N. Syrbu, I.G. Stamov, A.I. Kamertsel, Sov. Phys. Semicond. 26 (1992) 665.
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